Question

Find :

$\int_{0}^{\frac{\pi}{4}}(\sqrt{tanx}+\sqrt{cotx})dx$​

Answer 1

Answer:

One way to do this is to use our knowledge of some Taylor series.

\begin{lgathered}\displaystyle e^x=1+x+\frac{x^2}{2!}+\cdots\\\\\Rightarrow xe^x = x+x^2+\frac{x^3}{2!}+\cdots\\\\\text{and}\\\\\log(1+x)=x-\frac{x^2}2+\frac{x^3}{3}-\cdots\end{lgathered}ex=1+x+2!x2​+⋯⇒xex=x+x2+2!x3​+⋯andlog(1+x)=x−2x2​+3x3​−⋯​

Thus

\begin{lgathered}\displaystyle xe^x-\log(1+x)=\tfrac32x^2+\tfrac16x^3+\cdots\\\\\Rightarrow\frac{xe^x-\log(1+x)}{x^2}=\tfrac32+\tfrac16x+\cdots\\\\\Rightarrow\lim_{x\rightarrow0}\frac{xe^x-\log(1+x)}{x^2}=\frac32\end{lgathered}