Question

Evaluate $\int \frac{1}{\cos hx+\sin hx}dx$.

Answer 1

$\int \frac{1}{\cos hx+\sin hx}dx$

As we know that,

$\cos hx=\frac{e^x+e^{-x}}{2}$

$\sin hx=\frac{e^x-e^{-x}}{2}$

Therefore,

$\int \frac{1}{\cos hx+\sin hx}dx$

$=\int \frac{1}{\left(\frac{e^x+e^{-x}}{2}\right)+\left(\frac{e^x-e^{-x}}{2}\right)}dx$

$=\int \frac{2}{2e^x}dx$

$=\int e^{-x}dx$

$=-e^{-x}+C$