$$
\int_0^{\pi/2} \frac{
\sqrt[3]{\tan x}
}{
\left( \sin x + \cos x \right)^2
}\, dx
$$
This question I use the Kings Property and I equal to cube root of $\tan x$ plus cube root of $\cot x$ divided by $(\sin x+\cos x)^2$ but was stuck after that.
Or if we pull out $\cos x$ from the denominator, then it will be $(\tan x+1)^2$ and then substituting $\tan x$ for $u$. I couldn’t proceed further…
Kindly help
Thanks
Best Answer
Substitute $t={\tan x}$ \begin{align} &\int_0^{\pi/2} \frac{ \sqrt[3]{\tan x} }{ \left( \sin x + \cos x \right)^2 }\, dx\\ =&\int_0^\infty \frac {t^{1/3}}{(1+t)^2}dt\overset{ibp} = \int_0^\infty \frac {d(t^{1/3})}{1+t}\overset{t^{1/3}\to t}= \int_0^\infty \frac {dt}{1+t^3}=\frac{2\pi}{3\sqrt3} \end{align}