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Integrals

Multiple Choice Questions

71.

The value of $\int_{- 2}^{2} (xcos (x) + \sin (x) + 1) d x$

72.

$\int_{π}^{16 π} |\sin (x)| d x$ is equal to

73.

$\int \frac{\cos (2 x)}{\cos (x)} dx$ is equal to

$2 \sin (x) + \log |\sec (x) + \tan (x)| + C$
$2 \sin (x) - \log |\sec (x) - \tan (x)| + C$
$2 \sin (x) - \log |\sec (x) + \tan (x)| + C$
$2 \sin (x) + \log |\sec (x) - \tan (x)| + C$

74.
$\int \frac{\sin^{8} (x) - \cos^{8} (x)}{1 - 2 \sin^{2} (x) \cos^{2} (x)} dx$

$- \frac{1}{2} \sin (2 x) + C$

$\frac{1}{2} \sin (2 x) + C$

$\frac{1}{2} \sin (x) + C$

$- \frac{1}{2} \sin (x) + C$

$- \frac{1}{2} \sin (2 x) + C$

Let I = $\int \frac{\sin^{8} (x) - \cos^{8} (x)}{1 - 2 \sin^{2} (x) \cos^{2} (x)} dx$

= $\int \frac{(\sin^{4} (x) + \cos^{4} (x)) (\sin^{2} (x) + \cos^{2} (x)) (\sin^{2} (x) - \cos^{2} (x))}{1 - 2 \sin^{2} (x) - 2 \cos^{2} (x)} dx$

= $\int \frac{[{(\sin^{2} (x) + \cos^{2} (x))}^{2} - 2 \sin^{2} (x) \cos^{2} (x)] (1) (\sin^{2} (x) \cos^{2} (x))}{1 - 2 \sin^{2} (x) - 2 \cos^{2} (x)} dx$

= $\int \frac{(\sin^{2} (x) \cos^{2} (x))}{1 - 2 \sin^{2} (x) - \cos^{2} (x)} dx$

= $\int (\sin^{2} (x) - \cos^{2} (x)) dx = - \int \cos (2 x) dx$

= $- \frac{\sin (2 x)}{2} + C$

75.

The value of $\int_{0}^{π} \sin^{50} (x) \cos^{49} (x) d x$ is

0
$\frac{π}{4}$
$\frac{π}{2}$
1

76.

$\int 2^{x} (f' (x) + f (x) \log (2)) dx$ is

2^xf'(x) + C
2^xf(x) + C
2^x(log(2))f(x) + C
log(2)f(x) + C

Short Answer Type

77.

Evaluate the following integral

$\int_{- 1}^{2} |xsin (πx)| d x$

Multiple Choice Questions

78.

$\int \frac{\log (\sqrt{x})}{3 x} dx$ is equal to

$\frac{1}{3} {(\log (\sqrt{x}))}^{2} + c$
$\frac{2}{3} {(\log (\sqrt{x}))}^{2} + c$
$\frac{2}{3} {(\log (x))}^{2} + c$
$\frac{1}{3} {(\log (x))}^{2} + c$

79.

$\int e^{x} (\frac{2}{x} - \frac{2}{x^{2}}) dx$

$\frac{e^{x}}{x} + c$
$\frac{e^{x}}{2 x^{2}} + c$
$\frac{2 e^{x}}{x} + c$
$\frac{2 e^{x}}{x^{2}} + c$

80.

The value of the integral $\int \frac{dx}{{(e^{x} + e^{- x})}^{2}}$

$\frac{1}{2} (e^{2 x} + 1) + c$
$\frac{1}{2} (e^{- 2 x} + 1) + c$
$- \frac{1}{2} {(e^{2 x} + 1)}^{- 1} + c$
$\frac{1}{4} (e^{2 x} - 1) + c$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

74. ∫sin8x - cos8x1 - 2sin2xcos2xdx - 12sin2x + C 12sin2x + C 12sinx + C - 12sinx + C

Short Answer Type

Multiple Choice Questions

74.
$\int \frac{\sin^{8} (x) - \cos^{8} (x)}{1 - 2 \sin^{2} (x) \cos^{2} (x)} dx$

$- \frac{1}{2} \sin (2 x) + C$

$\frac{1}{2} \sin (2 x) + C$

$\frac{1}{2} \sin (x) + C$

$- \frac{1}{2} \sin (x) + C$