Chapter 5 of 8

63%

NDA Mathematics★ 12% of NDA Mathematics — roughly 14–15 of 120 questions, worth close to 35–38 of 300 marks

Integral Calculus & Differential Equations — NDA Mathematics

NDA integral calculus is a closed toolkit — a dozen standard forms, three techniques (substitution, parts, partial fractions), six properties of definite integrals, and first-order-first-degree differential equations solved by separating variables. No JEE-style trig substitutions, no second-order DEs. Learn the forms cold, know which technique a given integral is begging for, and never read off a DE's degree before clearing the radicals.

⏱ 24 min readmedium-hard difficulty✓ Free · exam-pattern aligned

Timed practiceMCQs from this chapter with instant scoring

FlashcardsFormulas & traps on spaced repetition

NDA PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Recall and apply the standard integral form table (algebraic, trig, exponential, and the six 'denominator' inverse-trig/log forms) to evaluate integrals by direct recognition
2Choose the correct technique — substitution, integration by parts (via ILATE), or partial fractions — for a given integral, including catching the 'numerator = derivative of denominator' log shortcut
3Evaluate definite integrals via the Fundamental Theorem and apply the six standard properties, including King's rule and the odd/even shortcuts, to convert a hard integral into a fast one
4Compute the area under a single curve or between two curves, including splitting at sign changes and applying the modulus correctly
5Determine the order and degree of a differential equation after rationalising radicals, and form the DE governing a given family of curves by eliminating arbitrary constants
6Solve first-order-first-degree differential equations by variable separation, including applied growth/decay problems using N = N₀e^(kt)

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Why this chapter matters in NDA

Integral Calculus & Differential Equations carries 12% of NDA Mathematics — the direct sequel to Differential Calculus (13%), and together the two calculus chapters account for a quarter of the entire paper. Unlike its parent topic, this chapter is a closed, learnable toolkit rather than an open-ended procedure: a dozen standard forms, three named techniques (substitution, by parts, partial fractions), six definite-integral properties, and one differential-equation technique (variable separation) cover essentially everything asked — NDA deliberately stays clear of trigonometric substitution, reduction formulae, and second-order differential equations. That narrowness is what makes it one of the fastest-scoring topics on the paper: a candidate who has the form table and the King's-rule trick reflexive can clear a differential-equation-heavy or definite-integral-heavy question in well under a minute, which matters enormously under the −0.8333 penalty in a 150-minute, 120-question, no-calculator paper.

Before you start — revise these

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Differential Calculus (NDA Mathematics)

Integration is defined as the reverse of differentiation — recognising which standard derivative a given integrand 'undoes' is the fastest route to the correct antiderivative, and derivative notation (y′, y″) carries straight into differential equations.

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Algebra & Trigonometry (NDA Mathematics)

Partial-fraction decomposition leans on factoring and solving for unknown constants, while several standard integral forms and definite-integral property questions (King's rule on sin/cos ratios) require fluent trig identities.

Integral Calculus & Differential Equations — NDA Mathematics

Differentiation asks "given a function, what is its rate of change?" Integration asks the reverse question, and NDA tests that reverse question in a narrow, learnable band: recognise a standard form, pick the right technique, and — for differential equations — separate the variables and integrate both sides. There is no trigonometric-substitution rabbit hole here. Master the form list and this becomes one of the fastest-scoring topics on the Mathematics paper.

1. What NDA actually asks

About 12% of the Mathematics paper — roughly 14–15 of the 120 questions, worth close to 35–38 marks. The syllabus is Class 11–12 CBSE-level and stays there; NDA does not venture into reduction formulae, Walli's formula, or higher-order differential equations. Six clusters cover essentially everything asked:

Cluster	What's tested	Typical count
Standard forms & substitution	Direct "evaluate this integral" using the form table or an obvious substitution	3–4 Q
Integration by parts / partial fractions	Products of functions, or rational functions that need decomposition	2–3 Q
Definite integrals — direct & property-based	Plug-and-evaluate, or a "shortcut via symmetry" question (odd/even, king's rule)	3–4 Q
Basic area under a curve	Area bounded by a line, parabola, or circle — never a multi-curve maze	1–2 Q
Order, degree & formation of a DE	Pure definition recall, or eliminate constants from a given family of curves	2 Q
Solving first-order-first-degree DEs	Variable-separable equation, sometimes dressed as growth/decay	2–3 Q

Every one of these is a recognition task, not a derivation task. The exam rewards a memorised form table and fast pattern-matching far more than clever manipulation.

2. Standard integral forms & techniques

The core table — know every entry without hesitation:

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1) \int \frac{1}{x} d x = ln ∣ x ∣ + C \int e^{x} d x = e^{x} + C \int a^{x} d x = \frac{a ^{x}}{l n a} + C$

$\int sin x d x = - cos x + C \int cos x d x = sin x + C \int sec^{2} x d x = tan x + C \int csc^{2} x d x = - cot x + C$

$\int sec x tan x d x = sec x + C \int csc x cot x d x = - csc x + C \int tan x d x = ln ∣ sec x ∣ + C \int cot x d x = ln ∣ sin x ∣ + C$

The "denominator" forms — these give inverse-trig or log answers, and NDA loves testing whether you can match a given denominator to the right one:

$\int \frac{d x}{a ^{2} - x ^{2}} = sin^{- 1} \frac{x}{a} + C \int \frac{d x}{a ^{2} + x ^{2}} = \frac{1}{a} tan^{- 1} \frac{x}{a} + C \int \frac{d x}{x x ^{2} - a ^{2}} = \frac{1}{a} sec^{- 1} \frac{x}{a} + C$

$\int \frac{d x}{a ^{2} - x ^{2}} = \frac{1}{2 a} ln \frac{a + x}{a - x} + C \int \frac{d x}{x ^{2} - a ^{2}} = \frac{1}{2 a} ln \frac{x - a}{x + a} + C \int \frac{d x}{x ^{2} \pm a ^{2}} = ln x + x^{2} \pm a^{2} + C$

Substitution is the workhorse: whenever the integrand contains a function and (a multiple of) its own derivative, substitute $u$ = that inner function. The instant giveaway is $\int f^{'} (x) g (f (x)) d x$ . A very common special case: if the numerator is exactly the derivative of the denominator, the answer is $ln ∣ denominator ∣ + C$ — no substitution even needs to be written out.

Integration by parts:

$\int u v d x = u \int v d x - \int (\frac{d u}{d x} \int v d x) d x$

Pick $u$ using ILATE (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential — in that priority order for which factor to treat as $u$ ). A useful bonus result that shows up as a "shortcut" question:

$\int e^{x} [f (x) + f^{'} (x)] d x = e^{x} f (x) + C$

Partial fractions handle rational functions $\frac{P ( x )}{Q ( x )}$ where $Q (x)$ factors into simple pieces. For distinct linear factors, write $\frac{1}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}$ , clear denominators, and find $A, B$ by plugging in $x = a$ and $x = b$ — no need to compare coefficients term by term. Repeated linear factors need an extra term for each power; an irreducible quadratic factor needs a linear numerator over it. Once decomposed, every piece integrates via the log or inverse-tan forms above.

3. Definite integrals & basic area applications

A definite integral evaluates by the Fundamental Theorem: $\int_{a}^{b} f (x) d x = F (b) - F (a)$ , where $F$ is any antiderivative of $f$ — no $+ C$ needed once limits are applied.

The six properties that turn a hard integral into a ten-second answer:

Property	Statement
P1	$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$
P2	$\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x$ , for $a < c < b$
P3 (King's rule)	$\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$
P4	$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ (P3 with $a = 0$ )
P5	$\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ if $f (2 a - x) = f (x)$ ; $= 0$ if $f (2 a - x) = - f (x)$
P6	$\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ if $f$ is even; $= 0$ if $f$ is odd

P6 is the fastest win in the entire chapter: spot an odd integrand on a symmetric interval and the answer is 0 without integrating a single term. P3/P4 (king's rule) is the tool behind the classic " $sin x$ over $sin x + cos x$ " family of integrals.

Area under a curve. If $f (x) \geq 0$ on $[a, b]$ , the area bounded by $y = f (x)$ , the $x$ -axis, and $x = a, x = b$ is $A = \int_{a}^{b} f (x) d x$ . If the curve dips below the axis, split at the zero and add the absolute value of the negative piece — a plain integral across a sign change under-counts the area. For the area between two curves with $f (x) \geq g (x)$ on $[a, b]$ : $A = \int_{a}^{b} [f (x) - g (x)] d x$ .

Two "basic" shapes NDA draws its area questions from:

Circle $x^{2} + y^{2} = a^{2}$ : Area $= 4 \int_{0}^{a} a^{2} - x^{2} d x = π a^{2}$ (the familiar formula, now derived).
Parabola $y^{2} = 4 a x$ cut off by its latus rectum $x = a$ : Area $= 2 \int_{0}^{a} 4 a x d x = \frac{8 a ^{2}}{3}$ .

4. Differential equations basics

A differential equation (DE) relates a function to its derivatives. Two definitions to fix permanently:

Order = the order of the highest derivative appearing in the equation.
Degree = the power of the highest-order derivative, after the equation has been made a polynomial in its derivatives (no fractional powers, no derivative under a radical or in a denominator). Squaring or rationalising to clear a radical is often the first — and most forgotten — step before degree can even be read off.

Formation of a differential equation. Given a family of curves with $n$ arbitrary constants, differentiate $n$ times and eliminate the constants algebraically; the result is an $n$ th-order DE satisfied by the entire family. Example: $y = A cos x + B sin x$ has two constants. Differentiating: $y^{'} = - A sin x + B cos x$ , and again: $y^{''} = - A cos x - B sin x = - y$ . Both constants vanish, leaving the second-order DE $y^{''} + y = 0$ — true for every member of the family, whatever $A$ and $B$ are.

General vs. particular solution. A solution containing the full quota of arbitrary constants (equal to the DE's order) is the general solution; substituting given initial/boundary values to pin down those constants gives the particular solution.

Solving by variable separation — the one technique NDA actually expects you to execute, not just recognise. If $\frac{d y}{d x} = f (x) g (y)$ , rewrite as $\frac{d y}{g ( y )} = f (x) d x$ and integrate both sides independently, adding a single constant of integration on one side.

Growth and decay is the applied face of the same equation. If a quantity's rate of change is proportional to its current value, $\frac{d N}{d t} = k N$ , separating and integrating gives $N = N_{0} e^{k t}$ — growth for $k > 0$ , decay for $k < 0$ . For radioactive decay with half-life $T$ , setting $N = N_{0} /2$ at $t = T$ gives the standard link $k = \frac{ln 2}{T}$ (taking $k$ as the magnitude of the decay constant).

Worked examples

Question 1 of 6

Q1. Evaluate $\int \frac{3 x ^{2}}{x ^{3} + 1} d x$ .

Show explanation

Solution. The numerator is exactly the derivative of the denominator, so the answer is a direct log form: $ln ∣ x^{3} + 1∣ + C$ .

Question 2 of 6

Q2. Evaluate $\int x e^{2 x} d x$ .

Show explanation

Solution. By parts with $u = x$ , $d v = e^{2 x} d x$ : $u = x \Rightarrow d u = d x$ ; $v = \frac{e ^{2 x}}{2}$ . $\int x e^{2 x} d x = \frac{x e ^{2 x}}{2} - \int \frac{e ^{2 x}}{2} d x = \frac{x e ^{2 x}}{2} - \frac{e ^{2 x}}{4} + C = \frac{e ^{2 x}}{4} (2 x - 1) + C$ (Check by differentiating: $\frac{d}{d x} [\frac{e ^{2 x}}{4} (2 x - 1)] = \frac{e ^{2 x}}{2} (2 x - 1) + \frac{e ^{2 x}}{2} = x e^{2 x}$ ✓.)

Question 3 of 6

Q3. Evaluate $\int_{0}^{π /2} \frac{sin x}{sin x + cos x} d x$ .

Show explanation

Solution. Let $I$ be this integral. Apply P4 ( $x \to \frac{π}{2} - x$ ): $sin (\frac{π}{2} - x) = cos x$ and $cos (\frac{π}{2} - x) = sin x$ , so $I = \int_{0}^{π /2} \frac{c o s x}{c o s x + s i n x} d x = J$ Adding: $I + J = \int_{0}^{π /2} 1 d x = \frac{π}{2}$ . Since $I = J$ , $2 I = \frac{π}{2}$ , so $I = \frac{π}{4}$ . (This king's-rule move — add the integral to its "flip" — is the single highest-value trick in the definite-integral section.)

Question 4 of 6

Q4. Find the area bounded by the parabola $y^{2} = 4 x$ and the line $x = 1$ .

Show explanation

Solution. Here $4 a = 4 \Rightarrow a = 1$ . Area $= \frac{8 a ^{2}}{3} = \frac{8}{3}$ sq. units. (Direct check: $2 \int_{0}^{1} 2 x d x = 4 [\frac{2}{3} x^{3/2}]_{0}^{1} = \frac{8}{3}$ ✓.)

Question 5 of 6

Q5. Solve $\frac{d y}{d x} = \frac{1 + y ^{2}}{1 + x ^{2}}$ , given $y = 1$ when $x = 0$ .

Show explanation

Solution. Separate: $\frac{d y}{1 + y ^{2}} = \frac{d x}{1 + x ^{2}} \Rightarrow tan^{- 1} y = tan^{- 1} x + C$ . At $x = 0, y = 1$ : $\frac{π}{4} = 0 + C \Rightarrow C = \frac{π}{4}$ . Particular solution: $tan^{- 1} y - tan^{- 1} x = \frac{π}{4}$ , which rearranges (tangent addition, $tan \frac{π}{4} = 1$ ) to $y = \frac{1 + x}{1 - x}$ .

Question 6 of 6

Q6. Find the differential equation of the family $y = A cos x + B sin x$ .

Show explanation

Solution. $y^{'} = - A sin x + B cos x$ ; $y^{''} = - A cos x - B sin x = - y$ . Both constants are eliminated: $y^{''} + y = 0$ (order 2, degree 1) — the DE of simple harmonic motion.

6. Common traps

Dropping the constant multiplier from substitution. $\int x^{2} e^{x^{3}} d x$ needs $u = x^{3}, d u = 3 x^{2} d x$ — the answer carries a $\frac{1}{3}$ , easy to lose under time pressure.
Reading degree before rationalising. A DE like $y^{''} = 1 + (y^{'})^{2}$ has degree 2, not 1 — you must square both sides first to clear the radical before the "power of the highest derivative" question even makes sense.
Sign slips in partial fractions. Plugging the wrong root into the wrong bracket flips a sign; always verify by adding the fractions back and checking the numerator matches.
Half-applying a definite-integral property. P5/P6 have conditions attached ( $f (2 a - x) = f (x)$ , or $f$ even/odd) — using the "shortcut" answer without checking the condition first is a guaranteed wrong option on the paper.
Forgetting to take the modulus in area problems. If the curve crosses the $x$ -axis inside $[a, b]$ , a single un-split integral lets positive and negative regions cancel, silently under-reporting the area.
Leaving an arbitrary constant in a "formed" DE. If the final equation still contains the original constant, the elimination is incomplete — the whole point of formation is a constant-free relation.
ILATE run backwards. Choosing the exponential or trig factor as $u$ in integration by parts usually regenerates a harder integral than the one you started with; algebraic and logarithmic factors almost always belong in $u$ first.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

Basic power/exponential/log forms

∫xⁿdx = xⁿ⁺¹/(n+1)+C (n≠−1) · ∫dx/x = ln|x|+C · ∫eˣdx = eˣ+C · ∫aˣdx = aˣ/ln a+C

The starting four — every polynomial or exponential integral reduces to these.

Standard trig forms

∫sin x dx=−cos x+C · ∫cos x dx=sin x+C · ∫sec²x dx=tan x+C · ∫csc²x dx=−cot x+C · ∫sec x tan x dx=sec x+C · ∫csc x cot x dx=−csc x+C · ∫tan x dx=ln|sec x|+C · ∫cot x dx=ln|sin x|+C

Know all eight cold — NDA tests direct recognition, not derivation.

Denominator (inverse-trig / log) forms

Match the given denominator's shape to the right form before doing any algebra — the single most-tested skill in the chapter.

Substitution — recognition rule

∫f′(x)·g(f(x)) dx: substitute u=f(x); special case ∫f′(x)/f(x) dx = ln|f(x)|+C

The giveaway is a function and (a multiple of) its own derivative sitting together in the integrand.

Integration by parts

∫u v dx = u∫v dx − ∫[(du/dx)∫v dx] dx, with u chosen by ILATE (Inverse trig, Log, Algebraic, Trig, Exponential)

Pick u as the earlier factor in ILATE order — algebraic/log factors almost always belong in u, not exponential/trig.

By-parts shortcut

∫eˣ[f(x)+f′(x)] dx = eˣf(x)+C

A pure recognition question — no actual by-parts computation needed once you spot the f(x)+f′(x) pairing.

Partial fractions — distinct linear factors

1/[(x−a)(x−b)] = A/(x−a) + B/(x−b); clear denominators and solve for A, B by substituting x=a and x=b

Every resulting piece then integrates via the log or inverse-tan forms above.

Fundamental Theorem of Calculus

∫ₐᵇ f(x)dx = F(b) − F(a), where F is any antiderivative of f

No +C once limits are applied — F(b)−F(a) is a fixed number, not a family.

Six properties of definite integrals

P1: ∫ₐᵇf=−∫ᵦᵃf · P2: ∫ₐᵇf=∫ₐᶜf+∫ᶜᵇf · P3: ∫ₐᵇf(x)dx=∫ₐᵇf(a+b−x)dx (King's rule) · P4: ∫₀ᵃf(x)dx=∫₀ᵃf(a−x)dx · P5: ∫₀²ᵃf=2∫₀ᵃf if f(2a−x)=f(x), =0 if f(2a−x)=−f(x) · P6: ∫₋ₐᵃf=2∫₀ᵃf if f even, =0 if f odd

P6 (odd/even) is the fastest win in the chapter; P3/P4 (King's rule) — add the integral to its own 'flipped' version — solves the whole sin/(sin+cos) family.

Area under a curve

Single curve (f≥0 on [a,b]): A=∫ₐᵇf(x)dx · Between curves (f≥g on [a,b]): A=∫ₐᵇ[f(x)−g(x)]dx · Circle x²+y²=a²: A=πa² · Parabola y²=4ax cut by x=a: A=8a²/3

If f dips below the axis inside [a,b], split at the zero and add the absolute value of the negative piece.

Order and degree of a DE

Order = order of the highest derivative present; Degree = power of the highest-order derivative once the equation is a polynomial in derivatives (radicals/fractions cleared first)

Rationalise (square, cube, etc.) before reading off the degree — the single most-tested conceptual trap.

Variable separation & growth/decay

dy/dx=f(x)g(y) ⟹ ∫dy/g(y) = ∫f(x)dx + C; growth/decay: dN/dt=kN ⟹ N=N₀eᵏᵗ, with k=(ln2)/T for half-life T

Add exactly one constant of integration, on one side only, after separating.

NDA test series

Ready to be tested on this?

Integral Calculus & Differential Equations — NDA Mathematics shows up in every NDA paper — sit a timed one and find out where you actually stand.

13–16Qs in paper

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Traps NDA sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Dropping the constant multiplier that substitution introduces

✓ ∫x²e^(x³)dx needs u=x³, du=3x²dx — the answer carries a 1/3 that's easy to lose under time pressure. After substituting, always check whether du matched the integrand exactly or only up to a constant.

WATCH OUT

✗ Reading a differential equation's degree before rationalising it

✓ A DE like y″=√(1+(y′)²) has degree 2, not 1 — square (or cube, etc.) both sides first to clear the radical before the 'power of the highest derivative' question even makes sense. Degree is only defined once the equation is a polynomial in its derivatives.

WATCH OUT

✗ Sign slips in partial fractions

✓ Plugging the wrong root into the wrong bracket flips a sign on A or B. Always verify by adding the fractions back together and checking the numerator reproduces the original.

WATCH OUT

✗ Half-applying a definite-integral property

✓ P5 and P6 have conditions attached (f(2a−x)=f(x), or f even/odd) — using the 'shortcut' answer without checking the condition first is a guaranteed wrong option. Verify f(−x) or f(2a−x) explicitly before applying the shortcut.

WATCH OUT

✗ Forgetting the modulus (absolute value) in area problems

✓ If the curve crosses the x-axis inside [a,b], a single un-split integral lets positive and negative regions cancel, silently under-reporting the area. Split at the zero, integrate each piece separately, and add the absolute values.

WATCH OUT

✗ Leaving an arbitrary constant in a 'formed' differential equation

✓ If the final equation still contains the original constant (e.g. dy/dx=Ay), the elimination is incomplete — the entire point of formation is a constant-free relation true for every member of the family.

WATCH OUT

✗ Running ILATE backwards in integration by parts

✓ Choosing the exponential or trig factor as u usually regenerates a harder integral than the one you started with. Algebraic and logarithmic factors almost always belong in u first — check the ILATE priority before picking.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Integral Calculus & Differential Equations — NDA Mathematics?

12 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

12 questions~8 min worth ~5 marks in NDA exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Know all eight standard trig integrals and the six 'denominator' forms (inverse-trig/log) cold — recognition, not derivation, is what's tested.
•Substitution giveaway: a function and (a multiple of) its own derivative sitting together; numerator = derivative of denominator ⇒ answer is ln|denominator|+C.
•Integration by parts: pick u via ILATE (Inverse trig, Log, Algebraic, Trig, Exponential); bonus shortcut ∫eˣ[f+f′]dx = eˣf(x)+C needs no computation at all.
•Partial fractions: 1/[(x−a)(x−b)]=A/(x−a)+B/(x−b), solve A and B by plugging in x=a and x=b — no coefficient-matching needed.
•Definite integrals need no +C — apply F(b)−F(a) directly once limits are in place.
•P6 (odd/even) is the fastest win: an odd f on [−a,a] gives 0 with zero integration. P3/P4 (King's rule): add I to its x→(a+b−x) flip to solve the whole sin/(sin+cos) family.
•Area under a curve: split at sign changes and add absolute values; a circle gives πa², and a parabola's latus-rectum area is 8a²/3.
•Degree of a DE is only defined after clearing radicals/fractions from the derivatives — rationalise first, then read the highest power.
•Formation of a DE: differentiate n times (n = number of arbitrary constants) and eliminate every constant algebraically — a leftover constant means the elimination isn't finished.
•Variable separable: dy/dx=f(x)g(y) ⟹ separate and integrate both sides, adding one constant total.
•Growth/decay: N=N₀eᵏᵗ; the half-life link is k=(ln2)/T.

NDA question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Typical weightage: ~14–15 questions × 2.5 marks ≈ 35–38 of 300 Mathematics marks (12% weight)

Question style	Marks each	Typical count	What it tests
Standard forms & substitution	2.5	3-4	Direct evaluation via the form table, or an obvious substitution including the numerator=derivative-of-denominator log shortcut
Integration by parts & partial fractions	2.5	2-3	Products of unlike functions (ILATE), and rational functions needing decomposition
Definite integrals — direct & property-based	2.5	3-4	Plug-and-evaluate via FTC, or a King's-rule/odd-even shortcut question
Area under a curve	2.5	1-2	Area bounded by a line, parabola, or circle — never a multi-curve maze
Order, degree & formation of a DE	2.5	2	Definition recall (rationalise before reading degree), or eliminating constants from a family of curves
Solving first-order-first-degree DEs	2.5	2-3	Variable-separable equations, sometimes dressed as growth/decay

Prep strategy

Week 1: write the full standard-integral form table (all eight trig forms + six denominator forms) from memory daily until it's reflexive — this chapter is close to half pure recall.
Week 2: drill technique-selection speed — given 20 mixed integrals, correctly identify substitution vs by-parts vs partial fractions in under 10 seconds each before actually solving.
Week 3: time yourself on the six definite-integral properties, especially King's rule (P3/P4) and the odd/even shortcut (P6); then drill order/degree-with-rationalisation and variable-separable DEs together, since both hinge on the same 'clear it first, then read/solve' discipline.
In the final week, mix all six sub-clusters into single practice sets (matching the real paper's spread) rather than drilling each technique in isolation — NDA questions arrive unlabelled and you need instant technique recognition, not just execution.

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Scan the integrand for a standard form first — a huge fraction of NDA integral questions are pure recognition against the form table, needing zero technique at all.
Before picking a technique, ask: is it a rational function (partial fractions), a product of unlike functions (by parts, via ILATE), or a function next to its own derivative (substitution)? This three-way check takes seconds and rarely misfires.
For definite integrals, always test the King's-rule / odd-even shortcuts (P3–P6) before integrating directly — verify the condition (f(2a−x)=±f(x), or f even/odd) explicitly rather than assuming it holds.
Never forget +C on an indefinite integral, and never carry a +C into a definite integral's final numeric answer — both are guaranteed-wrong-option traps.
For any differential equation, rationalise first (clear radicals/fractions in the derivatives) before stating order or degree — and for 'formation' questions, keep differentiating and eliminating until every arbitrary constant is gone.
No calculator is allowed — keep partial-fraction and King's-rule algebra clean by verifying each intermediate step (e.g. adding fractions back together) rather than rushing to a final answer.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Ballistics and motion analysis

Displacement is the integral of velocity and velocity the integral of acceleration — the reverse of the differentiation chain used in projectile and fire-control calculations, letting a trajectory be reconstructed from a recorded velocity or acceleration profile.

Radioactive decay and population modelling

The variable-separable equation dN/dt=kN and its solution N=N₀e^(kt) — worked out by hand in Section 4 — is the exact model used for radioactive decay dating, bacterial growth, and epidemic-spread curves.

Engineering quantity and volume estimation

Computing the material needed for a component, the volume of an irregular tank, or the cross-sectional area of a terrain profile all reduce to the same 'area under a curve' definite integral taught here, just with a more complex boundary function.

Probability distributions

In probability theory, the area under a probability density curve over an interval gives the probability of an outcome in that range — the identical geometric idea as 'area bounded by a curve and the x-axis' from Section 3, just applied to a density function instead of a physical curve.

Where else this topic is tested

Prepare once, score in every exam that asks it.

CDS (Combined Defence Services) Elementary MathematicsVery high — near-identical syllabus and question style

AFCAT (technical/numerical sections)Medium — lighter treatment, mostly standard forms and basic definite integrals

CUET MathematicsHigh — same NCERT-level content and question style

JEE Main (Integral Calculus & Differential Equations)Conceptual overlap — same core forms plus trig-substitution and higher-order DEs at much greater depth

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Questions aspirants ask

Pulled from the Q&A community and mentor sessions.

It's noticeably narrower. NDA sticks to the dozen-odd standard forms, substitution, by-parts, and simple partial fractions — there's no trigonometric-substitution toolkit beyond the direct standard forms, no reduction formulae, and differential equations stop at first-order-first-degree variable-separable equations (no second-order DEs, no linear-DE integrating-factor method). Depth is Class 11–12 CBSE; the challenge is speed and zero calculator access, not extra theory.

Check the integrand's shape in this order: if it's a rational function P(x)/Q(x) with Q(x) factorable, it's partial fractions. If it's a product of two different function types (algebraic × trig, algebraic × exponential, etc.), it's by parts — use ILATE to pick u. If you spot a function sitting next to (a multiple of) its own derivative, it's substitution — this catches the majority of 'standard form' questions in under five seconds.

Only when the condition actually holds — check f(−x) or f(2a−x) before committing. When it does hold (an odd integrand on a symmetric interval, or a King's-rule-style sin/cos ratio), the property turns a multi-step integral into a one-line answer. Applying it without checking the condition first is one of the most common wrong answers on the paper.

No — NDA's differential-equations syllabus stops at forming a DE from a family of curves (which can produce a second-order DE as the *result*, like y″+y=0) and at solving first-order-first-degree equations by variable separation. You are never asked to *solve* a second-order DE.

Losing a constant multiplier during substitution — writing ∫x²e^(x³)dx as e^(x³)+C instead of (1/3)e^(x³)+C. It's an easy slip because the substitution feels 'done' once you've matched the form, and the 1/3 from du=3x²dx gets forgotten in the rush to write the answer.