Q.1 Evaluate ∫(2x + 3) dx.
x^2 + 3x + C
x^2 + 6x + C
2x^2 + 3x + C
x^2 + 9x + C
Explanation - Integral of 2x is x^2 and integral of 3 is 3x. Add constant C.
Correct answer is: x^2 + 3x + C
Q.2 Evaluate ∫ x^2 dx.
x^3 + C
x^3 / 3 + C
2x^3 + C
x^2 / 2 + C
Explanation - Integral of x^n is x^(n+1)/(n+1) + C. Here n=2, so x^3/3 + C.
Correct answer is: x^3 / 3 + C
Q.3 Evaluate ∫ e^x dx.
e^x + C
x e^x + C
ln|x| + C
1/e^x + C
Explanation - The integral of e^x with respect to x is e^x + C.
Correct answer is: e^x + C
Q.4 Evaluate ∫ (1/x) dx.
ln|x| + C
1/x^2 + C
x + C
e^x + C
Explanation - Integral of 1/x is natural log of absolute x, ln|x| + C.
Correct answer is: ln|x| + C
Q.5 Evaluate ∫ sin(x) dx.
-cos(x) + C
cos(x) + C
sin(x) + C
-sin(x) + C
Explanation - Integral of sin(x) is -cos(x) + C.
Correct answer is: -cos(x) + C
Q.6 Evaluate ∫ cos(x) dx.
sin(x) + C
-sin(x) + C
cos(x) + C
-cos(x) + C
Explanation - Integral of cos(x) is sin(x) + C.
Correct answer is: sin(x) + C
Q.7 Evaluate ∫ sec^2(x) dx.
tan(x) + C
cot(x) + C
sec(x) + C
-tan(x) + C
Explanation - Integral of sec^2(x) is tan(x) + C.
Correct answer is: tan(x) + C
Q.8 Evaluate ∫ csc^2(x) dx.
-cot(x) + C
cot(x) + C
csc(x) + C
-csc(x) + C
Explanation - Integral of csc^2(x) is -cot(x) + C.
Correct answer is: -cot(x) + C
Q.9 Evaluate ∫ dx / (1 + x^2).
tan^-1(x) + C
cot^-1(x) + C
ln|1+x^2| + C
x + C
Explanation - Integral of 1/(1 + x^2) is arctangent of x + C.
Correct answer is: tan^-1(x) + C
Q.10 Evaluate ∫ dx / √(1 - x^2).
sin^-1(x) + C
cos^-1(x) + C
sec^-1(x) + C
tan^-1(x) + C
Explanation - Integral of 1/√(1-x^2) is arcsin(x) + C.
Correct answer is: sin^-1(x) + C
Q.11 Evaluate ∫ x e^x dx.
x e^x - e^x + C
x e^x + e^x + C
e^x - x e^x + C
x^2 e^x + C
Explanation - Use integration by parts: ∫ u dv = uv - ∫ v du. u=x, dv=e^x dx.
Correct answer is: x e^x - e^x + C
Q.12 Evaluate ∫ ln(x) dx.
x ln(x) - x + C
x ln(x) + C
ln(x)/x + C
1/x + C
Explanation - Use integration by parts: ∫ ln(x) dx = x ln(x) - ∫ x * (1/x) dx = x ln(x) - x + C.
Correct answer is: x ln(x) - x + C
Q.13 Evaluate ∫ x^2 e^x dx.
x^2 e^x - 2x e^x + 2 e^x + C
x^2 e^x + 2x e^x + 2 e^x + C
x^2 e^x - x e^x + C
x^2 e^x + C
Explanation - Use integration by parts twice. Formula: ∫ x^n e^x dx = x^n e^x - n ∫ x^(n-1) e^x dx.
Correct answer is: x^2 e^x - 2x e^x + 2 e^x + C
Q.14 Evaluate ∫ cos^2(x) dx.
x/2 + sin(2x)/4 + C
x/2 - sin(2x)/4 + C
sin^2(x)/2 + C
x + C
Explanation - Use cos^2(x) = (1 + cos2x)/2. Integrate to get x/2 + sin2x/4 + C.
Correct answer is: x/2 + sin(2x)/4 + C
Q.15 Evaluate ∫ sin^2(x) dx.
x/2 - sin(2x)/4 + C
x/2 + sin(2x)/4 + C
sin^2(x)/2 + C
cos^2(x)/2 + C
Explanation - Use sin^2(x) = (1 - cos2x)/2. Integrate to get x/2 - sin2x/4 + C.
Correct answer is: x/2 - sin(2x)/4 + C
Q.16 Evaluate ∫ sec(x) dx.
ln|sec(x) + tan(x)| + C
ln|sec(x) - tan(x)| + C
tan(x) + C
sec(x) + C
Explanation - Standard integral: ∫ sec(x) dx = ln|sec(x)+tan(x)| + C.
Correct answer is: ln|sec(x) + tan(x)| + C
Q.17 Evaluate ∫ csc(x) dx.
ln|csc(x) - cot(x)| + C
ln|csc(x) + cot(x)| + C
cot(x) + C
csc(x) + C
Explanation - Standard integral: ∫ csc(x) dx = ln|csc(x) + cot(x)| + C.
Correct answer is: ln|csc(x) + cot(x)| + C
Q.18 Evaluate ∫ x / √(1 - x^2) dx.
-√(1 - x^2) + C
√(1 - x^2) + C
1 - x^2 + C
ln|1 - x^2| + C
Explanation - Use substitution u = 1 - x^2, du = -2x dx. Integral becomes -√(1 - x^2) + C.
Correct answer is: -√(1 - x^2) + C
Q.19 Evaluate ∫ 1 / (x^2 + a^2) dx.
1/a tan^-1(x/a) + C
1/a cot^-1(x/a) + C
ln|x^2 + a^2| + C
x / (x^2 + a^2) + C
Explanation - Standard integral: ∫ dx/(x^2 + a^2) = 1/a * arctan(x/a) + C.
Correct answer is: 1/a tan^-1(x/a) + C
Q.20 Evaluate ∫ dx / (x√(x^2 - a^2)).
1/a sec^-1(|x|/a) + C
1/a cos^-1(x/a) + C
ln|x + √(x^2 - a^2)| + C
1/√(x^2 - a^2) + C
Explanation - Standard integral: ∫ dx/(x√(x^2 - a^2)) = 1/a sec^-1(|x|/a) + C.
Correct answer is: 1/a sec^-1(|x|/a) + C
Q.21 Evaluate ∫ x / (x^2 + 1) dx.
1/2 ln|x^2 + 1| + C
ln|x^2 + 1| + C
x / (x^2 + 1) + C
1/(x^2 + 1) + C
Explanation - Substitute u = x^2 + 1, du = 2x dx. Integral becomes 1/2 ln|x^2 + 1| + C.
Correct answer is: 1/2 ln|x^2 + 1| + C
Q.22 Evaluate ∫ x^3 dx.
x^4 / 4 + C
x^3 / 3 + C
x^4 + C
3x^4 + C
Explanation - Integral of x^n is x^(n+1)/(n+1) + C. Here n=3, so x^4/4 + C.
Correct answer is: x^4 / 4 + C