Question

Solve the integral of tanhxsech

Answer 1

$$\int \tanh x\text{sech} xdx=\int\dfrac{\sinh x}{\cosh^2 x}dx$$

$$u=\cosh x, d=\sinh xdx$$

$$\int \tanh x\text{sech} xdx=\int\dfrac{\sinh x}{\cosh^2 x}dx$$

$$=\int\dfrac{du}{ u^2}=-\dfrac{1}{u}+C=-\dfrac{1}{\cosh x}+C=$$

$$=-\text{sech} x$$