Question

integrate; integrate it by substitution

Answer 1

hope this will help you

Answer 2

Solution :

Now, $\int \frac{dx}{e^{x}+e^{-x}}$

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

⇤ -- --- --- --- --- --- -- ⇥

Put $e^{x}=z$

⇢ $e^{x}\:dx=dz$

⇤ -- --- --- --- --- --- -- ⇥

= $\int \frac{dz}{z^{2}+1}$

= $tan^{-1}z+C$ ,

where C is constant of integration

= $tan^{-1}(e^{x})+C$ ,

which is the required integral.