Evaluate the definite integral: integral from -1 to 1 of (x+1) dx.
Hint. Find the antiderivative x^2/2+x, then apply the Second Fundamental Theorem: evaluate at the upper limit minus the lower limit.
The antiderivative is x^2/2+x. At x=1 this is 1/2+1=3/2, and at x=-1 this is 1/2-1=-1/2, so the definite integral is 3/2-(-1/2).
✦ 2
Evaluate the definite integral: integral from 2 to 3 of (1/x) dx.
Hint. The antiderivative of 1/x is log|x|; evaluate at both limits and subtract.
The antiderivative is log x (positive on this interval). Evaluating gives log3-log2, which combines into a single log by the quotient rule for logarithms.
✦ log(3/2)
Evaluate the definite integral: integral from 1 to 2 of (4x^3-5x^2+6x+9) dx.
Hint. Integrate term by term to get x^4-(5/3)x^3+3x^2+9x, then evaluate at both limits.
The antiderivative is x^4-(5/3)x^3+3x^2+9x. Evaluating at x=2 and x=1 and subtracting gives the final numeric value.
✦ 64/3
Evaluate the definite integral: integral from 0 to pi/4 of sin 2x dx.
Hint. The antiderivative is -(1/2)cos2x; evaluate at both limits.
The antiderivative is -(1/2)cos2x. At x=pi/4, cos(pi/2)=0, and at x=0, cos0=1, so the definite integral is 0-(-1/2).
✦ 1/2
Evaluate the definite integral: integral from 0 to pi/2 of cos 2x dx.
Hint. The antiderivative is (1/2)sin2x; evaluate at both limits.
The antiderivative is (1/2)sin2x. At x=pi/2, sin(pi)=0, and at x=0, sin0=0, so the definite integral is exactly zero.
✦ 0
Evaluate the definite integral: integral from 4 to 5 of e^x dx.
Hint. The antiderivative of e^x is e^x itself; evaluate at both limits.
Since e^x is its own antiderivative, evaluating at the two limits and subtracting gives e^5-e^4 directly.
✦ e^5 - e^4
Evaluate the definite integral: integral from 0 to pi/4 of tan x dx.
Hint. The antiderivative of tanx is log|secx|; evaluate at both limits.
The antiderivative is log|secx|. At x=pi/4, sec(pi/4)=sqrt(2), and at x=0, sec0=1, so the definite integral is log(sqrt(2))-log(1)=log(sqrt(2)).
✦ (1/2) log 2
Evaluate the definite integral: integral from pi/6 to pi/4 of cosec x dx.
Hint. The antiderivative of cosecx is log|tan(x/2)|; evaluate at both limits.
The antiderivative is log|tan(x/2)|. Evaluating at x=pi/4 (giving tan(pi/8)) and x=pi/6 (giving tan(pi/12)) and subtracting gives a difference of two logs.
✦ log(tan(pi/8)) - log(tan(pi/12))
Evaluate the definite integral: integral from 0 to 1 of dx/sqrt(1-x^2).
Hint. The antiderivative is sin^-1x; evaluate at both limits.
The antiderivative is sin^-1x. At x=1, sin^-1(1)=pi/2, and at x=0, sin^-1(0)=0, so the definite integral is pi/2-0.
✦ pi/2
Evaluate the definite integral: integral from 0 to 1 of dx/(1+x^2).
Hint. The antiderivative is tan^-1x; evaluate at both limits.
The antiderivative is tan^-1x. At x=1, tan^-1(1)=pi/4, and at x=0, tan^-1(0)=0, so the definite integral is pi/4-0.
✦ pi/4
Evaluate the definite integral: integral from 2 to 3 of dx/(x^2-1).
Hint. The antiderivative is (1/2)log|(x-1)/(x+1)|; evaluate at both limits.
The antiderivative is (1/2)log|(x-1)/(x+1)|. Evaluating at x=3 (giving log(1/2)) and x=2 (giving log(1/3)) and subtracting simplifies to a single log.
✦ (1/2) log(3/2)
Evaluate the definite integral: integral from 0 to pi/2 of cos^2 x dx.
Hint. Rewrite cos^2x=(1+cos2x)/2 first, then integrate term by term.
Using cos^2x=(1+cos2x)/2, the antiderivative is x/2+(1/4)sin2x. At x=pi/2, sin(pi)=0, and at x=0, everything vanishes, leaving just the x/2 term.
✦ pi/4
Evaluate the definite integral: integral from 2 to 3 of x/(x^2+1) dx.
Hint. The antiderivative is (1/2)log(x^2+1); evaluate at both limits.
With t=x^2+1, the antiderivative is (1/2)log(x^2+1). At x=3 this is (1/2)log10, and at x=2 this is (1/2)log5, so the difference is (1/2)log2.
✦ (1/2) log 2
Evaluate the definite integral: integral from 0 to 1 of (2x+3)/(5x^2+1) dx.
Hint. Split the fraction into a piece matching the derivative of 5x^2+1 and a standard tan^-1 piece, then evaluate.
Writing 2x+3 as (2/5)(10x) + 3, the integral splits into (1/5)log(5x^2+1) plus a tan^-1 term; evaluating both pieces at x=1 and x=0 and adding gives the numeric result.
✦ (1/5) log 6 + (3/sqrt(5)) tan^-1(sqrt(5))
Evaluate the definite integral: integral from 0 to 1 of x e^(x^2) dx.
Hint. Put t=x^2, so dt=2x dx, then evaluate the resulting exponential integral at the transformed limits.
With t=x^2, the antiderivative in x is (1/2)e^(x^2). At x=1 this is e/2, and at x=0 this is 1/2, so the definite integral is (e-1)/2.
✦ (e-1)/2
Evaluate the definite integral: integral from 1 to 2 of 5x^2/(x^2+4x+3) dx.
Hint. This fraction is improper — divide first, then decompose the proper remainder into partial fractions before integrating.
Dividing out the improper part and decomposing the remainder over (x+1)(x+3) gives antiderivative 5x+(5/2)log|x+1|-(45/2)log|x+3|; evaluating at x=2 and x=1 and subtracting gives the numeric result.
✦ 5 + (5/2) log(3/2) - (45/2) log(5/4)
Evaluate the definite integral: integral from 0 to pi/4 of (2sec^2 x+x^3+2) dx.
Hint. Integrate each of the three terms separately, then evaluate at both limits.
Term by term, the antiderivative is 2tanx+x^4/4+2x. At x=pi/4, tan(pi/4)=1, giving 2 plus the polynomial terms evaluated at pi/4; at x=0 everything vanishes.
✦ 2 + pi^4/1024 + pi/2
Evaluate the definite integral: integral from 0 to pi of (sin^2(x/2) - cos^2(x/2)) dx.
Hint. Recognise sin^2(x/2)-cos^2(x/2) = -cos x, then integrate.
Since sin^2(x/2)-cos^2(x/2) = -cosx (a standard double-angle identity), the antiderivative is -sinx, and sin(pi)=sin(0)=0, so the definite integral vanishes entirely.
✦ 0
Evaluate the definite integral: integral from 0 to 2 of (6x+3)/(x^2+4) dx.
Hint. Split into a piece matching the derivative of x^2+4 and a standard tan^-1 piece, then evaluate.
Writing 6x+3 as 3(2x)+3, the integral splits into 3log(x^2+4) plus (3/2)tan^-1(x/2); evaluating both pieces at x=2 and x=0 and adding gives the numeric result.
✦ 3 log 2 + 3pi/8 (equivalently, log 8 + 3pi/8)
Evaluate the definite integral: integral from 0 to 1 of (x e^x + sin(pi x/4)) dx.
Hint. Integrate the two terms separately — the first by parts, the second directly.
The first term x e^x integrates by parts to x e^x-e^x, giving 1 when evaluated from 0 to 1. The second term sin(pi x/4) integrates to -(4/pi)cos(pi x/4), giving (4-2sqrt(2))/pi when evaluated. Adding these gives the total.
✦ 1 + (4 - 2sqrt(2))/pi
Choose the correct answer: integral from 1 to sqrt(3) of dx/(1+x^2) equals (A) pi/3 (B) 2pi/3 (C) pi/6 (D) pi/12
Hint. The antiderivative is tan^-1x; evaluate at both limits and subtract.
The antiderivative is tan^-1x. At x=sqrt(3), tan^-1(sqrt(3))=pi/3, and at x=1, tan^-1(1)=pi/4, so the definite integral is pi/3-pi/4=pi/12.
✦ (D) pi/12
Choose the correct answer: integral from 0 to 2/3 of dx/(4+9x^2) equals (A) pi/6 (B) pi/12 (C) pi/24 (D) pi/4
Hint. Factor out 9 from the denominator to match the standard tan^-1 form, then evaluate at both limits.
Writing 4+9x^2=9(x^2+4/9), the antiderivative is (1/6)tan^-1(3x/2). At x=2/3, this gives (1/6)tan^-1(1)=(1/6)(pi/4)=pi/24, and at x=0 it is zero.
✦ (C) pi/24
Solutions written by the tuition.in editorial team and checked against lemh201.pdf (NCERT, Reprint 2026-27). Questions are referenced from the NCERT textbook for identification.
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