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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.

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

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

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

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

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

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

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

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

$I = \int \frac{\log (\sqrt{x})}{3 x} dx Put x = t^{2} \Rightarrow dx = 2 tdt ∴ I = \int \frac{\log (t)}{3 t^{2}} . 2 t dt = \int \frac{2}{3} . \frac{\log (t)}{t} dt Put \log (t) = z \Rightarrow \frac{1}{t} dt = dz ∴ I = \int \frac{2}{3} zdz = \frac{2}{3} . \frac{z^{2}}{2} + c = \frac{z^{2}}{3} + c = \frac{{(\log (t))}^{2}}{3} + c = \frac{\log {(\sqrt{x})}^{3}}{3} + c$