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Integrals

Multiple Choice Questions

181.

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

$\log |\frac{\sin (x)}{1 + \cos (x)}| + c$
$\log |\frac{\sin (x)}{x + \cos (x)}| + c$
$\log |\frac{2 \sin (x)}{x + \cos (x)}| + c$
$\log |\frac{x}{x + \cos (x)}| + c$

182.
The integral $\int_{0}^{1} \frac{2 \sin^{- 1} (\frac{x}{2})}{x} dx$ equals

$\int_{0}^{\frac{π}{6}} \frac{xdx}{\tan (x)}$

$\int_{0}^{\frac{π}{6}} \frac{2}{\tan (x)} dx$

$\int_{0}^{\frac{π}{2}} \frac{2 xdx}{\tan (x)}$

$\int_{0}^{\frac{π}{6}} \frac{2 x}{\sin (x)} dx$

$\int_{0}^{\frac{π}{6}} \frac{2}{\tan (x)} dx$

$z = \int_{0}^{1} \frac{2 \sin^{- 1} (\frac{x}{2})}{x} dx Put \sin^{- 1} (\frac{x}{2}) = t \Rightarrow x = 2 \sin (t) \Rightarrow dx = 2 \cos (t) dt Also, x = 0 \Rightarrow t = 0 x = 1 \Rightarrow t = \frac{π}{6} ∴ I = \int_{0}^{\frac{π}{6}} \frac{2 t}{2 \sin (t)} . 2 \cos (t) dt = \int_{0}^{\frac{π}{6}} \frac{2 t}{\tan (t)} dt = \int_{0}^{\frac{π}{6}} \frac{2 x}{\tan (x)} dx$