First slide

Evaluation of definite integrals

Question

If $\int_{0}^{2} \frac{d x}{\sqrt{x + 1} + \sqrt{(x + 1)^{3}}}$ is equal to $k π$ then $k$ is equal to

Easy

A

$\frac{1}{2}$
B

2
C

$\frac{1}{6}$
D

$\frac{1}{3}$

Solution

The given integral in equal to
$\int_{0}^{2} \frac{d x}{\sqrt{x + 1} (2 + x)} = \int_{1}^{\sqrt{3}} \frac{2 t d t}{t (t^{2} + 1)}, t^{2} = x + 1$
$\begin{aligned} = 2 \int_{1}^{\sqrt{3}} \frac{d t}{t^{2} + 1} = {2 \tan^{- 1} t|}_{1}^{\sqrt{3}} \\ = 2 [\frac{π}{3} - \frac{π}{4}] = \frac{π}{6} . \end{aligned}$

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