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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-07-CT
Integrals — Full Chapter Test
Chapter: Integrals
Topic: Complete chapter — all topics
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • This is a full-length test covering the whole chapter — every topic is included.
  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
$\displaystyle\int x^3\,dx=$
  • A.$3x^2+C$
  • B.$\tfrac{x^4}{4}+C$
  • C.$\tfrac{x^4}{3}+C$
  • D.$x^4+C$
2.
$\displaystyle\int_0^2 x^2\,dx=$
  • A.$\tfrac{8}{3}$
  • B.$4$
  • C.$8$
  • D.$\tfrac{4}{3}$
3.
$\displaystyle\int \tfrac1x\,dx=$
  • A.$\ln|x|+C$
  • B.$-\tfrac{1}{x^2}+C$
  • C.$x\ln x+C$
  • D.$1+C$
4.
$\displaystyle\int_{-a}^{a} f(x)\,dx=0$ when $f$ is:
  • A.even
  • B.odd
  • C.positive
  • D.constant
5.
$\displaystyle\int 2x\,e^{x^2}\,dx=$
  • A.$e^{x^2}+C$
  • B.$x^2e^{x^2}+C$
  • C.$2e^{x^2}+C$
  • D.$e^{2x}+C$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Evaluate $\displaystyle\int (3x^2+2x)\,dx$.
7.
Evaluate $\displaystyle\int_0^1 x\,dx$.
8.
Evaluate $\displaystyle\int \cos x\,dx$.
9.
Evaluate $\displaystyle\int_0^{\pi/2}\cos x\,dx$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Evaluate $\displaystyle\int x e^x\,dx$.
11.
Evaluate $\displaystyle\int_0^1 (3x^2+1)\,dx$.
12.
Evaluate $\displaystyle\int \tfrac{1}{x^2-1}\,dx$.
13.
Evaluate $\displaystyle\int_1^3 (2x+1)\,dx$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Evaluate $\displaystyle\int \tfrac{x^2+1}{x+1}\,dx$.
15.
Evaluate $\displaystyle\int_0^{\pi/2}\sin^2 x\,dx$.

Answer Key

Section A — Multiple Choice Questions
  1. (B) $\tfrac{x^4}{4}+C$
  2. (A) $\tfrac{8}{3}$
  3. (A) $\ln|x|+C$
  4. (B) odd
  5. (A) $e^{x^2}+C$
Section B — Short Answer (2 marks)
  1. $x^3+x^2+C$.
  2. $\tfrac12$.
  3. $\sin x+C$.
  4. $1$.
Section C — Short Answer (3 marks)
  1. $e^x(x-1)+C$.
  2. $2$.
  3. $\tfrac12\ln\left|\tfrac{x-1}{x+1}\right|+C$.
  4. $10$.
Section D — Long Answer (5 marks)
  1. $\tfrac{x^2}{2}-x+2\ln|x+1|+C$.
  2. $\tfrac{\pi}{4}$.
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