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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-07-DEF-01
Definite Integrals & Properties — Assignment
Chapter: Integrals
Topic: Definite Integrals & Their Properties
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • 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.
$\displaystyle\int_0^2 x^2\,dx=$
  • A.$\tfrac{8}{3}$
  • B.$4$
  • C.$8$
  • D.$\tfrac{4}{3}$
2.
$\displaystyle\int_{-a}^{a} f(x)\,dx=0$ when $f$ is:
  • A.even
  • B.odd
  • C.positive
  • D.constant
3.
$\displaystyle\int_0^{\pi}\sin x\,dx=$
  • A.$0$
  • B.$1$
  • C.$2$
  • D.$\pi$
4.
$\displaystyle\int_a^a f(x)\,dx=$
  • A.$f(a)$
  • B.$0$
  • C.$1$
  • D.$2f(a)$
5.
The Fundamental Theorem gives $\displaystyle\int_a^b f\,dx=$
  • A.$F(a)-F(b)$
  • B.$F(b)-F(a)$
  • C.$F(b)+F(a)$
  • D.$f(b)-f(a)$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Evaluate $\displaystyle\int_0^1 x\,dx$.
7.
Evaluate $\displaystyle\int_0^{\pi/2}\cos x\,dx$.
8.
Evaluate $\displaystyle\int_1^2 2x\,dx$.
9.
Evaluate $\displaystyle\int_{-2}^{2} x^3\,dx$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Evaluate $\displaystyle\int_0^1 (3x^2+1)\,dx$.
11.
Evaluate $\displaystyle\int_1^3 (2x+1)\,dx$.
12.
Evaluate $\displaystyle\int_{-1}^{1} (x^3+x^2)\,dx$.
13.
Evaluate $\displaystyle\int_0^{\pi/2}\tfrac{\sin x}{\sin x+\cos x}\,dx$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Evaluate $\displaystyle\int_0^{\pi/2}\sin^2 x\,dx$.
15.
Evaluate $\displaystyle\int_0^{\pi}\tfrac{x}{1+\sin x}\,dx$ using $\int_0^a f(x)dx=\int_0^a f(a-x)dx$.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $\tfrac{8}{3}$
  2. (B) odd
  3. (C) $2$
  4. (B) $0$
  5. (B) $F(b)-F(a)$
Section B — Short Answer (2 marks)
  1. $\tfrac12$.
  2. $1$.
  3. $3$.
  4. $0$.
Section C — Short Answer (3 marks)
  1. $2$.
  2. $10$.
  3. $\tfrac23$.
  4. $\tfrac{\pi}{4}$.
Section D — Long Answer (5 marks)
  1. $\tfrac{\pi}{4}$.
  2. $\pi$.
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