Question

$\int_{}^{}\frac{dx}{4\sin^{2}x + 5\cos^{2}x}$

• A. $\frac{1}{\sqrt{5}}\tan^{- 1}\left( \frac{2\tan x}{\sqrt{5}} \right) + c$
• B. $\frac{1}{\sqrt{5}}\tan^{- 1}\left( \frac{\tan x}{\sqrt{5}} \right) + c$
• C. $\frac{1}{2\sqrt{5}}\tan^{- 1}\left( \frac{2\tan x}{\sqrt{5}} \right) + c$
• D. None of these

Answer 1

Answer: (C)

$\frac{1}{2\sqrt{5}}\tan^{- 1}\left( \frac{2\tan x}{\sqrt{5}} \right) + c$

Answer 2

$\int \frac { \sec ^ { 2 } x d x } { 4 \tan ^ { 2 } x + 5 }$ $\left( \begin{aligned} & \text{Put}2\tan x = t \\ & 2\sec^{2}xdx = dt \end{aligned} \right)$

$\Rightarrow \frac{1}{2}\int_{}^{}\frac{dt}{t^{2} + (\sqrt{5})^{2}} = \frac{1}{2\sqrt{5}}\tan^{- 1}\left( \frac{2\tan x}{\sqrt{5}} \right) + c$