Question

$\int_{}^{}\frac{\mathbf{dx}}{\mathbf{(2}\mathbf{\sin}\mathbf{x}\mathbf{+}\mathbf{\cos}\mathbf{x}\mathbf{)}^{\mathbf{2}}}$

• A. $\frac{1}{2}\left( \frac{1}{2\tan x + 1} \right)$
• B. $\frac{1}{2}\log(2\tan x + 1) + c$
• C. $\frac{1}{2 + \cot x} + c$
• D. $- \frac{1}{2}\left( \frac{1}{2\tan x - 1} \right) + c$

Answer 1

Answer: (C)

$\frac{1}{2 + \cot x} + c$

Answer 2

$\int_{}^{}{\frac{dx}{(2\sin x + \cos x)^{2}} = \int_{}^{}\frac{dx}{\sin^{2}x(2 + \cot x)^{2}}}$ $= \int_{}^{}\frac{\cos ⥂ ec^{2}xdx}{(2 + \cot x)^{2}}$

Put $(2 + \cot x) = t \Rightarrow - \cos ⥂ ec^{2}xdx = dt$

$= \int_{}^{}{\frac{- dt}{t^{2}} = \frac{1}{t} + c}$ $= \frac{1}{2 + \cot x} + c$