Integrals Test

Integrals

This is Integrals Test-03 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
\(\int {|x|dx} \)
  • A
    0
  • B
    \(\frac{{{x^2}}}{2}\)
  • C
    \(\frac{{x|x|}}{2}\)
    Correct
  • D
    \( - \frac{{{x^2}}}{2}\)
2
\(\int {(\sin (\log x) + \cos (\log x))} \) dx is equal to
  • A
    sin (log x) – cos (log x)
  • B
    x cos (log x)
  • C
    none of these
  • D
    x sin (log x)
    Correct
3
\(\int {{{\tan }^{ - 1}}x} \)dx is equal to
  • A
    \(x{\tan ^{ - 1}}x + \frac{1}{2}\log (1 + {x^2})\)
  • B
    none of these
  • C
    \(x{\tan ^{ - 1}}x + \log (1 + {x^2})\)
  • D
    \(x{\tan ^{ - 1}}x - \frac{1}{2}\log (1 + {x^2})\)
    Correct
4
\(\int\limits_a^b {\frac{{\log x}}{x}dx} \) is equal to
  • A
    \(\frac{1}{2}\log (ab)\log \left( {\frac{b}{a}} \right)\)
    Correct
  • B
    log (a + b) log ( b –a)
  • C
    log (a b) log \(\left( {\frac{b}{a}} \right)\)
  • D
    none of these
5
If \(\int\limits_{ - 2}^5 {f(x)dx = 4,\int\limits_0^5 {(1 + f(x))dx = 7,} } \) then the value of the integral \(\int\limits_{ - 2}^0 {f(x)} \) dx is equal to
  • A
    – 3
  • B
    5
  • C
    3
  • D
    2
    Correct
6
If f (x) = f (a –x), then , \(\int\limits_0^a {xf(x)dx} \) is equal to
  • A
    \(\frac{{{a^2}}}{2}\int\limits_0^a {f(x)dx} \)
  • B
    \(\int\limits_0^a {f(x)dx} \)
  • C
    \(\frac{a}{2}\int\limits_0^a {f(x)dx} \)
    Correct
  • D
    \( - \frac{{{a^2}}}{2}\int\limits_0^a {f(x)dx} \)
7
\(\int {\frac{{\cos 4x + 1}}{{\cot x + \tan x}}dx} \) is equal to
  • A
    \( - \frac{1}{6}{\sin ^3}2x + C\)
  • B
    \(\frac{1}{6}{\cos ^3}2x + C\)
  • C
    \( - \frac{1}{6}{\cos ^3}2x + C\)
    Correct
  • D
    none of these.
8
\(\int\limits_0^\pi {\sqrt {1 - \cos x} } \)is equal to
  • A
    2
  • B
    None of these
  • C
    \(2\sqrt 2 \)
    Correct
  • D
    1
9
\(\int\limits_0^\pi {\sqrt {1 + \cos x} } \) dx is equal to
  • A
    2
  • B
    \(2\sqrt 2 \)
    Correct
  • C
    1
  • D
    \(\sqrt 2 \)
10
\(\int\limits_0^{\pi /2} {\log (\sin x)} \)dx
  • A
    \(\pi \;log{\text{ }}2\)
  • B
    \(\frac{\pi }{2}\log 2\)
  • C
    \(--\pi \;log{\text{ }}2\)
  • D
    \( - \frac{\pi }{2}\log 2\)
    Correct
11
\(\int\limits_0^1 {{e^ - }^{{{\sin }^2}x}} \)dx is equal to
  • A
    – 1
  • B
    none of these
    Correct
  • C
    2
  • D
    \(1 + \frac{1}{e}\)
12
If \(\int\limits_0^a {{x^m}} {(a - x)^n}dx\)\( = \lambda \int\limits_0^1 {{x^n}{{(a - x)}^m}} \)dx, then \(\lambda \) is equal
  • A
    none of these
  • B
    1
    Correct
  • C
    \(\frac{1}{2}\)
  • D
    – 1
13
\(\int\limits_0^{\pi /2} {\sin x} \) sin 2 x dx is equal to
  • A
    none of these.
  • B
    \(\frac{2}{3}\)
    Correct
  • C
    \(\frac{\pi }{3}\)
  • D
    \(\frac{\pi }{3}\)
14
\(\int\limits_0^\pi {\sqrt {1 + \sin x} } \)dx is equal to
  • A
    2
  • B
    none of these
  • C
    \(2\sqrt 2 \)
  • D
    4
    Correct
15
\(\int\limits_0^1 {\frac{{1 - x}}{{1 + x}}} \;\) dx is equal to
  • A
    2 log 2 + 1
  • B
    2 log 2 – 1
    Correct
  • C
    \(\sqrt 2 \log 2 - 1\)
  • D
    1 – 2 log 2