Q.1 What is the integral of x with respect to x?
x^2/2 + C
2x + C
ln(x) + C
1/x + C
Explanation - The power rule of integration states ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1. Here n = 1, so ∫x dx = x^2/2 + C.
Correct answer is: x^2/2 + C
Q.2 ∫ cos(x) dx = ?
sin(x) + C
-sin(x) + C
cos^2(x)/2 + C
-cos(x) + C
Explanation - The derivative of sin(x) is cos(x). Hence, the integral of cos(x) is sin(x) + C.
Correct answer is: sin(x) + C
Q.3 Find ∫ e^x dx.
e^x + C
ln(x) + C
x·e^x + C
1/e^x + C
Explanation - The derivative of e^x is e^x itself, so the integral of e^x is e^x + C.
Correct answer is: e^x + C
Q.4 What is ∫ 1/x dx for x > 0?
ln(x) + C
1/(x^2) + C
x + C
e^x + C
Explanation - The derivative of ln(x) is 1/x. Therefore, ∫(1/x) dx = ln(x) + C.
Correct answer is: ln(x) + C
Q.5 Evaluate ∫ 2x dx.
x^2 + C
2x^2 + C
ln(x) + C
2 + C
Explanation - Using the power rule, ∫2x dx = (2/2)x^2 = x^2 + C.
Correct answer is: x^2 + C
Q.6 The integral of sin(x) dx is:
-cos(x) + C
cos(x) + C
sin^2(x)/2 + C
-sin(x) + C
Explanation - The derivative of -cos(x) is sin(x), hence ∫sin(x) dx = -cos(x) + C.
Correct answer is: -cos(x) + C
Q.7 Evaluate ∫ (3x^2) dx.
x^3 + C
3x^3 + C
x^2 + C
9x^2 + C
Explanation - Using the power rule: ∫3x^2 dx = (3/3)x^3 = x^3 + C.
Correct answer is: x^3 + C
Q.8 What is ∫ 0 dx?
0
C
x + C
Undefined
Explanation - The integral of 0 with respect to x is a constant, denoted as C.
Correct answer is: C
Q.9 ∫ sec^2(x) dx = ?
tan(x) + C
-tan(x) + C
sec(x) + C
-sec(x) + C
Explanation - The derivative of tan(x) is sec^2(x), hence ∫sec^2(x) dx = tan(x) + C.
Correct answer is: tan(x) + C
Q.10 ∫ 5 dx = ?
5x + C
x^5 + C
ln(5) + C
C
Explanation - The integral of a constant a with respect to x is ax + C.
Correct answer is: 5x + C
Q.11 Evaluate ∫ (x^3) dx.
x^4/4 + C
4x^3 + C
3x^2 + C
ln(x) + C
Explanation - Using the power rule: ∫x^3 dx = x^(3+1)/(3+1) = x^4/4 + C.
Correct answer is: x^4/4 + C
Q.12 ∫ dx = ?
x + C
1/x + C
ln(x) + C
C
Explanation - The integral of 1 with respect to x is x + C.
Correct answer is: x + C
Q.13 Find ∫ cos(2x) dx.
(1/2)sin(2x) + C
2sin(2x) + C
-(1/2)cos(2x) + C
tan(2x) + C
Explanation - Let u = 2x. Then ∫cos(2x) dx = (1/2)sin(2x) + C.
Correct answer is: (1/2)sin(2x) + C
Q.14 What is ∫ x^-1 dx?
ln|x| + C
x + C
1/x^2 + C
x^-2/(-2) + C
Explanation - For n = -1, ∫x^n dx = ln|x| + C.
Correct answer is: ln|x| + C
Q.15 Evaluate ∫ tan(x) dx.
ln|sec(x)| + C
-ln|cos(x)| + C
sec(x) + C
sin(x) + C
Explanation - ∫tan(x) dx = ∫(sin(x)/cos(x)) dx = -ln|cos(x)| + C.
Correct answer is: -ln|cos(x)| + C
Q.16 ∫ (4x^3 + 2) dx = ?
x^4 + 2x + C
x^4/4 + 2x + C
x^4 + x^2 + C
2x^4 + 2x + C
Explanation - Integrate each term: ∫4x^3 dx = x^4, ∫2 dx = 2x. So result is x^4 + 2x + C.
Correct answer is: x^4 + 2x + C
Q.17 ∫ e^(2x) dx = ?
(1/2)e^(2x) + C
2e^(2x) + C
e^(2x) + C
ln(e^(2x)) + C
Explanation - Let u = 2x, then du = 2 dx. ∫e^(2x) dx = (1/2)e^(2x) + C.
Correct answer is: (1/2)e^(2x) + C
Q.18 ∫ ln(x) dx = ?
x ln(x) - x + C
ln(x)^2/2 + C
1/x + C
x^2 ln(x) + C
Explanation - Using integration by parts: ∫ln(x) dx = x ln(x) - x + C.
Correct answer is: x ln(x) - x + C
Q.19 Find ∫ 1/(1+x^2) dx.
arctan(x) + C
ln(1+x^2) + C
arcsin(x) + C
sqrt(1+x^2) + C
Explanation - The derivative of arctan(x) is 1/(1+x^2). Hence ∫1/(1+x^2) dx = arctan(x) + C.
Correct answer is: arctan(x) + C
Q.20 Evaluate ∫ sinh(x) dx.
cosh(x) + C
sinh(x) + C
tanh(x) + C
ln(cosh(x)) + C
Explanation - The derivative of cosh(x) is sinh(x). Hence ∫sinh(x) dx = cosh(x) + C.
Correct answer is: cosh(x) + C
Q.21 ∫ cosh(x) dx = ?
sinh(x) + C
cosh(x) + C
tanh(x) + C
ln(sinh(x)) + C
Explanation - The derivative of sinh(x) is cosh(x). Hence ∫cosh(x) dx = sinh(x) + C.
Correct answer is: sinh(x) + C
Q.22 Evaluate ∫ (1/sqrt(1-x^2)) dx.
arcsin(x) + C
arccos(x) + C
arctan(x) + C
ln|x| + C
Explanation - The derivative of arcsin(x) is 1/sqrt(1-x^2). Therefore, ∫1/sqrt(1-x^2) dx = arcsin(x) + C.
Correct answer is: arcsin(x) + C
Q.23 ∫ x e^x dx = ?
x e^x - e^x + C
x e^x + C
e^x + C
x^2 e^x + C
Explanation - Using integration by parts: let u = x, dv = e^x dx. Then result = x e^x - ∫e^x dx = x e^x - e^x + C.
Correct answer is: x e^x - e^x + C
Q.24 Find ∫ x cos(x) dx.
x sin(x) + cos(x) + C
x sin(x) - cos(x) + C
sin(x) - x cos(x) + C
cos(x) - x sin(x) + C
Explanation - Integration by parts: u = x, dv = cos(x) dx → du = dx, v = sin(x). So result = x sin(x) - ∫sin(x) dx = x sin(x) + cos(x) + C.
Correct answer is: x sin(x) + cos(x) + C