First slide

Methods of integration

Question

The value of integral $\int \frac{d x}{\cos^{3} x \sqrt{\sin 2 x}}$ can be expressed as

Moderate

A

a rational function of tan x
B

an irrational function tan x
C

a rational function of cos x
D

a rational function of sin x

Solution

write the given integral as
$I = \frac{1}{\sqrt{2}} \int \frac{d x}{\cos^{4} x \sqrt{\tan x}}$
put $\tan x = t$ to obtain
$\begin{array}{l} I = \frac{1}{\sqrt{2}} \int \frac{1 + t^{2}}{\sqrt{t}} d t = \frac{1}{\sqrt{2}} \int [t^{- 1 / 2} + t^{3 / 2}] d t \\ = \frac{1}{\sqrt{2}} [2 \sqrt{t} + \frac{2}{5} t^{5 / 2}] + C \\ = \sqrt{2} [\sqrt{\tan x} + \frac{1}{5} (\tan x)^{5 / 2}] + C . \end{array}$
Thus, $I$ is an irrational function of $\tan x$

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