Q01
MCQ
The integral of e^x dx is:
(a) e^x + C
(b) xe^x + C
(c) e^(x+1) + C
(d) log x + C (1 mark)
CBSE · Class 12 · Mathematics
Integrals quiz
Q02
MCQ
The integral of 1/x dx (x not 0) is:
(a) x^2 + C
(b) log abs(x) + C
(c) -1/x^2 + C
(d) e^x + C (1 mark)
Q03
MCQ
The integral of sec^2 x dx is:
(a) tan x + C
(b) sec x + C
(c) cot x + C
(d) sec x tan x + C (1 mark)
Q04
MCQ
The integral of x dx from 0 to 1 is:
(a) 1
(b) 1/2
(c) 2
(d) 0 (1 mark)
Q05
MCQ
The integral of x^3 dx from -1 to 1 is:
(a) 1/2
(b) 1/4
(c) 0
(d) 2 (1 mark)
Q06
MCQ
The integral of 1/sqrt(1 - x^2) dx is:
(a) cos^-1 x + C
(b) sin^-1 x + C
(c) tan^-1 x + C
(d) log x + C (1 mark)
Q07
MCQ
The integral of sin x dx from 0 to pi/2 is:
(a) 0
(b) 1
(c) -1
(d) pi/2 (1 mark)
Q08
MCQ
The integral of 1/(1 + x^2) dx from 0 to 1 is:
(a) pi/2
(b) pi/4
(c) 1
(d) pi (1 mark)
Q09
MCQ
The integral of e^x (f(x) + f'(x)) dx is:
(a) e^x f'(x) + C
(b) e^x f(x) + C
(c) f(x) + C
(d) e^x + C (1 mark)
Q10
MCQ
The integral of cos x dx is:
(a) -sin x + C
(b) sin x + C
(c) cos x + C
(d) -cos x + C (1 mark)
Q11
Short
Evaluate the integral of tan^2 x dx. (2 marks)
Q12
Short
Evaluate the integral of (log x)/x dx. (2 marks)
Q13
Short
Evaluate the integral of 1/(x^2 - 9) dx. (2 marks)
Q14
Short
Evaluate the integral of log x dx. (2 marks)
Q15
Short
Evaluate the integral of (x^2 + 1) dx from 0 to 2. (2 marks)
Q16
Short
Evaluate the integral of 1/(x log x) dx. (2 marks)
Q17
Short
Evaluate the integral of sin^5 x dx from -pi/2 to pi/2, giving a reason. (2 marks)
Q18
Short
Evaluate the integral of x^2 e^(x^3) dx. (2 marks)
Q19
Short
Evaluate the integral of abs(x - 2) dx from 0 to 4. (2 marks)
Q20
Short
Evaluate the integral of sin x cos x dx. (2 marks)
Q21
Numerical
Evaluate the integral of tan x dx from 0 to pi/4. (3 marks)
Q22
Numerical
Evaluate the integral of x e^x dx from 0 to 1. (3 marks)
Q23
Numerical
Evaluate the integral of x/(x^2 + 1) dx from 2 to 3. (3 marks)
Q24
Numerical
Evaluate the integral of x sqrt(2 - x) dx from 0 to 2. (3 marks)
Q25
Numerical
Evaluate the integral of (sin x - cos x)/(1 + sin x cos x) dx from 0 to pi/2. (3 marks)
Q26
Case
Read the passage and answer the questions. The marginal cost of producing x units of a product is MC = 2x + 5 rupees, and the fixed cost is Rs 100. (i) Find the total cost function C(x). (ii) Find the total cost of producing 10 units. (iii) Find the increase in cost when output rises from 10 to 20 units. (5 marks)
Q27
Case
Read the passage and answer the questions. A car starts from rest and its velocity after t seconds is v(t) = 3t^2 + 2t m/s. (i) Write the integral for distance covered in the first 2 seconds. (ii) Evaluate it. (iii) Find the distance covered in the third second. (5 marks)
Q28
Case
Read the passage and answer the questions. A student needs to evaluate the integral of 1/(x^2 - 4) dx from 3 to 4. (i) Split the integrand into partial fractions. (ii) Find the indefinite integral. (iii) Evaluate the definite integral. (5 marks)
Q29
Case
Read the passage and answer the questions. Let I be the integral of sin x/(sin x + cos x) dx from 0 to pi/2. A teacher suggests replacing x by (pi/2 - x). (i) Write I after the substitution. (ii) Add the two forms of I. (iii) Find I. (5 marks)
Q30
Case
Read the passage and answer the questions. To evaluate the integral of x^2 e^x dx, a student uses integration by parts twice. (i) Which function is taken as the first function and why? (ii) Carry out the first step. (iii) Find the final answer. (5 marks)
Q31
Long/Diagram
Evaluate the integral of log(1 + tan x) dx from 0 to pi/4. (6 marks)
Q32
Long/Diagram
Evaluate the integral of 1/(x(x^2 + 1)) dx using partial fractions. (6 marks)
Q33
Long/Diagram
Evaluate the integral of (abs(x - 1) + abs(x - 2) + abs(x - 3)) dx from 1 to 4, sketching the graph of the integrand. (6 marks)
Q34
Long/Diagram
Evaluate the integral of e^(2x) sin x dx. (6 marks)
Q35
Long/Diagram
Evaluate the integral of x/(1 + sin x) dx from 0 to pi. (6 marks)
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