0-pi-4-tanh-2x-dx-

Question Number 87861 by Rio Michael last updated on 06/Apr/20

\underset{0}{\int^{\frac{π}{4}}} \tanh 2 x d x

Commented by MJS last updated on 06/Apr/20

\tanh 2 x = \frac{e^{4 x} - 1}{e^{4 x} + 1}

\int \tanh 2 x d x =

[t = e^{4 x} \to d x = \frac{d t}{{4 e}^{4 x}}]

= \frac{1}{4} \int \frac{t - 1}{t (t + 1)} d t = \frac{1}{2} \int \frac{d t}{t + 1} - \frac{1}{4} \int \frac{d t}{t} =

= \frac{1}{2} \ln (t + 1) - \frac{1}{4} \ln t = - x + \frac{1}{2} \ln (e^{4 x} + 1) + C

Commented by Rio Michael last updated on 06/Apr/20

sir can we use^{″} integration by {parts}^{″} ?

Commented by MJS last updated on 07/Apr/20

I {don}^{'} t think that would be easier

Commented by Rio Michael last updated on 07/Apr/20

i will try it out sir