Integrate: sqrt(4-x^2).
Hint. Apply the standard formula for integral of sqrt(a^2-x^2) with a=2.
This matches the standard form integral of sqrt(a^2-x^2)dx = (x/2)sqrt(a^2-x^2)+(a^2/2)sin^-1(x/a) directly with a=2.
✦ (x/2) sqrt(4-x^2) + 2 sin^-1(x/2) + C
Integrate: sqrt(1-4x^2).
Hint. Put u=2x to reduce this to the standard form sqrt(1-u^2), then rescale for du=2dx.
With u=2x, du=2dx, the integral becomes (1/2) integral of sqrt(1-u^2)du; since this matches the standard form (u/2)sqrt(1-u^2)+(1/2)sin^-1(u), replacing u back by 2x gives the result.
✦ (x/2) sqrt(1-4x^2) + (1/4) sin^-1(2x) + C
Integrate: sqrt(x^2+4x+6).
Hint. Complete the square: x^2+4x+6=(x+2)^2+2, matching the standard form sqrt(u^2+a^2).
Completing the square gives (x+2)^2+2, so the integral matches the standard form (u/2)sqrt(u^2+a^2)+(a^2/2)log|u+sqrt(u^2+a^2)| with u=x+2, a^2=2.
✦ ((x+2)/2) sqrt(x^2+4x+6) + log|x+2+sqrt(x^2+4x+6)| + C
Integrate: sqrt(x^2+4x+1).
Hint. Complete the square: x^2+4x+1=(x+2)^2-3, matching the standard form sqrt(u^2-a^2).
Completing the square gives (x+2)^2-3, so the integral matches the standard form (u/2)sqrt(u^2-a^2)-(a^2/2)log|u+sqrt(u^2-a^2)| with u=x+2, a^2=3.
✦ ((x+2)/2) sqrt(x^2+4x+1) - (3/2) log|x+2+sqrt(x^2+4x+1)| + C
Integrate: sqrt(1-4x-x^2).
Hint. Complete the square: 1-4x-x^2 = 5-(x+2)^2, matching the standard form sqrt(a^2-u^2).
Completing the square gives 5-(x+2)^2, so the integral matches the standard form (u/2)sqrt(a^2-u^2)+(a^2/2)sin^-1(u/a) with u=x+2, a^2=5.
✦ ((x+2)/2) sqrt(1-4x-x^2) + (5/2) sin^-1((x+2)/sqrt(5)) + C
Integrate: sqrt(x^2+4x-5).
Hint. Complete the square: x^2+4x-5=(x+2)^2-9, matching the standard form sqrt(u^2-a^2).
Completing the square gives (x+2)^2-9, so the integral matches the standard form (u/2)sqrt(u^2-a^2)-(a^2/2)log|u+sqrt(u^2-a^2)| with u=x+2, a=3.
✦ ((x+2)/2) sqrt(x^2+4x-5) - (9/2) log|x+2+sqrt(x^2+4x-5)| + C
Integrate: sqrt(1+3x-x^2).
Hint. Complete the square: 1+3x-x^2 = 13/4-(x-3/2)^2, matching the standard form sqrt(a^2-u^2).
Completing the square gives 13/4-(x-3/2)^2, so the integral matches the standard form (u/2)sqrt(a^2-u^2)+(a^2/2)sin^-1(u/a) with u=x-3/2, a^2=13/4.
✦ ((2x-3)/4) sqrt(1+3x-x^2) + (13/8) sin^-1((2x-3)/sqrt(13)) + C
Integrate: sqrt(x^2+3x).
Hint. Complete the square: x^2+3x=(x+3/2)^2-9/4, matching the standard form sqrt(u^2-a^2).
Completing the square gives (x+3/2)^2-9/4, so the integral matches the standard form (u/2)sqrt(u^2-a^2)-(a^2/2)log|u+sqrt(u^2-a^2)| with u=x+3/2, a=3/2.
✦ ((2x+3)/4) sqrt(x^2+3x) - (9/8) log|x+3/2+sqrt(x^2+3x)| + C
Integrate: sqrt(1+x^2/9).
Hint. Factor out 1/9 from inside the root first: 1+x^2/9=(9+x^2)/9.
Since sqrt(1+x^2/9)=(1/3)sqrt(9+x^2), the integral matches the standard form (u/2)sqrt(u^2+a^2)+(a^2/2)log|u+sqrt(u^2+a^2)| with u=x, a=3, scaled by 1/3.
✦ (x/6) sqrt(9+x^2) + (3/2) log|x+sqrt(9+x^2)| + C
Choose the correct answer: integral of sqrt(1+x^2) dx is equal to (A) (x/2)sqrt(1+x^2)+(1/2)log|x+sqrt(1+x^2)|+C (B) (2/3)(1+x^2)^(3/2)+C (C) (2/3)x(1+x^2)^(3/2)+C (D) (x^2/2)sqrt(1+x^2)+(1/2)x^2log|x+sqrt(1+x^2)|+C
Hint. Apply the standard formula for integral of sqrt(x^2+a^2) with a=1.
This matches the standard form (x/2)sqrt(x^2+a^2)+(a^2/2)log|x+sqrt(x^2+a^2)| directly with a=1.
✦ (A) (x/2) sqrt(1+x^2) + (1/2) log|x+sqrt(1+x^2)| + C
Choose the correct answer: integral of sqrt(x^2-8x+7) dx is equal to (A) (1/2)(x-4)sqrt(x^2-8x+7)+9log|x-4+sqrt(x^2-8x+7)|+C (B) (1/2)(x+4)sqrt(x^2-8x+7)+9log|x+4+sqrt(x^2-8x+7)|+C (C) (1/2)(x-4)sqrt(x^2-8x+7)-3sqrt(2)log|x-4+sqrt(x^2-8x+7)|+C (D) (1/2)(x-4)sqrt(x^2-8x+7)-(9/2)log|x-4+sqrt(x^2-8x+7)|+C
Hint. Complete the square: x^2-8x+7=(x-4)^2-9, matching the standard form sqrt(u^2-a^2).
Completing the square gives (x-4)^2-9, so the integral matches the standard form (u/2)sqrt(u^2-a^2)-(a^2/2)log|u+sqrt(u^2-a^2)| with u=x-4, a=3, so a^2/2=9/2.
✦ (D) (1/2)(x-4) sqrt(x^2-8x+7) - (9/2) log|x-4+sqrt(x^2-8x+7)| + C
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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