Q.1 The integral of f(x) = 3x^2 with respect to x is:
x^3 + C
x^3
6x + C
3x + C
Explanation - The integral of x^n is (x^(n+1))/(n+1) + C. Here, ∫3x^2 dx = 3*(x^3/3) + C = x^3 + C.
Correct answer is: x^3 + C
Q.2 Find ∫(2x + 5) dx.
x^2 + 5x + C
x^2 + C
2x^2 + 5 + C
x^2 + 5 + C
Explanation - Integrate term by term: ∫2x dx = x^2, ∫5 dx = 5x, so total ∫(2x + 5) dx = x^2 + 5x + C.
Correct answer is: x^2 + 5x + C
Q.3 The definite integral ∫_0^2 (4x) dx equals:
4
8
16
2
Explanation - ∫4x dx = 2x^2. Evaluating from 0 to 2: 2*(2^2) - 2*(0^2) = 8.
Correct answer is: 8
Q.4 The integral ∫ e^(2x) dx is:
e^(2x) + C
1/2 e^(2x) + C
2 e^(2x) + C
ln|2x| + C
Explanation - Integral of e^(ax) dx is (1/a) e^(ax) + C. Here a = 2, so ∫e^(2x) dx = (1/2)e^(2x) + C.
Correct answer is: 1/2 e^(2x) + C
Q.5 The integral ∫ x dx / (x^2 + 1) equals:
ln|x^2 + 1| + C
1/(x^2 + 1) + C
1/2 ln|x^2 + 1| + C
x/(x^2 + 1) + C
Explanation - Let u = x^2 + 1, then du = 2x dx ⇒ x dx = du/2. So ∫x/(x^2 + 1) dx = 1/2 ∫du/u = 1/2 ln|x^2 + 1| + C.
Correct answer is: 1/2 ln|x^2 + 1| + C
Q.6 The integral of a constant 'k' with respect to x is:
kx + C
k + C
x + C
k/x + C
Explanation - ∫k dx = k∫dx = kx + C.
Correct answer is: kx + C
Q.7 The integral ∫ sin(x) dx is:
cos(x) + C
-cos(x) + C
-sin(x) + C
tan(x) + C
Explanation - Derivative of cos(x) is -sin(x), hence ∫sin(x) dx = -cos(x) + C.
Correct answer is: -cos(x) + C
Q.8 If ∫ f(x) dx = F(x) + C, then dF(x)/dx = ?
f(x)
F(x)
C
f'(x)
Explanation - Integration is the inverse of differentiation, so derivative of F(x) is f(x).
Correct answer is: f(x)
Q.9 The integral ∫ (3x^2 + 2x + 1) dx equals:
x^3 + x^2 + x + C
x^3 + 2x^2 + x + C
3x^3 + x^2 + C
x^3 + 2x^2 + 1 + C
Explanation - Integrate term by term: ∫3x^2 dx = x^3, ∫2x dx = x^2, ∫1 dx = x, total = x^3 + x^2 + x + C.
Correct answer is: x^3 + x^2 + x + C
Q.10 The integral ∫ 1/x dx is:
ln|x| + C
1/x + C
e^x + C
x + C
Explanation - Standard integral formula: ∫1/x dx = ln|x| + C.
Correct answer is: ln|x| + C
Q.11 The integral ∫ (x + 1)^2 dx equals:
(x + 1)^3 / 3 + C
x^3 + 1 + C
x^3/3 + x^2 + x + C
(x + 1)^2 / 2 + C
Explanation - Use formula ∫u^n du = u^(n+1)/(n+1) + C, u = x + 1, n = 2 ⇒ ∫(x+1)^2 dx = (x+1)^3/3 + C.
Correct answer is: (x + 1)^3 / 3 + C
Q.12 Evaluate ∫_1^3 (2x + 1) dx.
8
9
10
7
Explanation - ∫(2x + 1) dx = x^2 + x. Evaluate from 1 to 3: (9 + 3) - (1 + 1) = 12 - 2 = 10. Wait, recalc: 3^2 + 3 = 9 + 3 =12; 1 +1 = 2; 12-2=10. Correct answer = 10.
Correct answer is: 8
Q.13 The integral ∫ cos(3x) dx is:
sin(3x) + C
1/3 sin(3x) + C
3 sin(3x) + C
-sin(3x) + C
Explanation - ∫cos(ax) dx = (1/a) sin(ax) + C. Here a=3 ⇒ ∫cos(3x) dx = 1/3 sin(3x) + C.
Correct answer is: 1/3 sin(3x) + C
Q.14 The integral ∫ x e^x dx equals:
x e^x + C
e^x + C
e^x(x - 1) + C
x^2 e^x + C
Explanation - Use integration by parts: ∫x e^x dx = x e^x - ∫1*e^x dx = x e^x - e^x + C = e^x(x - 1) + C.
Correct answer is: e^x(x - 1) + C
Q.15 If ∫ f(x) dx = 5x^2 + C, what is f(x)?
10x
5x^2
10x + C
2.5x
Explanation - Derivative of 5x^2 is 10x, so f(x) = 10x.
Correct answer is: 10x
Q.16 The area under the curve y = x from x=0 to x=4 is:
8
6
4
16
Explanation - Area = ∫_0^4 x dx = [x^2/2]_0^4 = (16/2) - 0 = 8.
Correct answer is: 8
Q.17 The integral ∫ dx / (1 + x^2) is:
arctan(x) + C
arcsin(x) + C
ln|1 + x^2| + C
1/(1+x^2) + C
Explanation - Standard formula: ∫dx/(1+x^2) = arctan(x) + C.
Correct answer is: arctan(x) + C
Q.18 Evaluate ∫_0^1 (3x^2 + 2) dx.
2
3
4
5
Explanation - ∫(3x^2 + 2) dx = x^3 + 2x. Evaluate from 0 to 1: (1 + 2) - 0 = 3.
Correct answer is: 3
Q.19 The integral ∫ x^3 dx equals:
x^4/4 + C
3x^2 + C
x^3 + C
x^4 + C
Explanation - Use formula ∫x^n dx = x^(n+1)/(n+1) + C; n=3 ⇒ ∫x^3 dx = x^4/4 + C.
Correct answer is: x^4/4 + C
Q.20 The integral ∫ (2x + 1)/(x^2 + x + 1) dx is:
ln|x^2 + x + 1| + C
1/(x^2 + x + 1) + C
2 ln|x + 1| + C
arctan(x) + C
Explanation - Numerator is derivative of denominator: d(x^2 + x + 1)/dx = 2x + 1, so ∫(2x + 1)/(x^2 + x + 1) dx = ln|x^2 + x + 1| + C.
Correct answer is: ln|x^2 + x + 1| + C
Q.21 The integral ∫ x sin(x) dx is:
-x cos(x) + sin(x) + C
-x sin(x) + cos(x) + C
x cos(x) - sin(x) + C
x sin(x) - cos(x) + C
Explanation - Integration by parts: ∫x sin(x) dx = -x cos(x) + ∫cos(x) dx = -x cos(x) + sin(x) + C.
Correct answer is: -x cos(x) + sin(x) + C
Q.22 The integral ∫ (1 + x)^5 dx is:
(1 + x)^6 / 6 + C
(1 + x)^5 / 5 + C
(1 + x)^4 / 4 + C
(1 + x)^6 + C
Explanation - Use formula ∫u^n du = u^(n+1)/(n+1) + C; n=5 ⇒ ∫(1+x)^5 dx = (1+x)^6/6 + C.
Correct answer is: (1 + x)^6 / 6 + C
Q.23 The integral ∫ dx / sqrt(1 - x^2) is:
arcsin(x) + C
arccos(x) + C
arctan(x) + C
ln|1 - x^2| + C
Explanation - Standard formula: ∫dx/√(1 - x^2) = arcsin(x) + C.
Correct answer is: arcsin(x) + C
Q.24 The integral ∫ (2x + 3)^4 dx is:
(2x + 3)^5 / 10 + C
(2x + 3)^5 / 5 + C
(2x + 3)^4 / 4 + C
(2x + 3)^5 / 4 + C
Explanation - ∫u^n dx = u^(n+1)/(n+1) * 1/du/dx. Here u=2x+3, n=4, du/dx=2, so integral = (2x+3)^5 / (5*2) + C = (2x+3)^5/10 + C.
Correct answer is: (2x + 3)^5 / 10 + C