Answer to Question #243086 in Calculus for Akki

Question #243086

Integral sech √x tanh √x/√x dx


1
Expert's answer
2021-09-28T10:02:54-0400

Let I=(sech(x).tanh(x)x)dxI=\int (\frac{sech(\sqrt{x}).tanh(\sqrt{x})}{\sqrt{x}}) dx\\

Put x=t\sqrt{x}=t

12xdx=dt1xdx=2.dt\therefore \frac{1}{2\sqrt{x}}dx=dt\\ \Rightarrow \frac{1}{\sqrt{x}}dx=2.dt

So, I=2(sech(t).tanh(t))dtI=2\int (sech(t).tanh(t))dt\\

=2(2et+et)(etetet+et)dt=4(etet(et+et)2)dt=2\int (\frac{2}{ e^t + e^{-t}})(\frac{ e^t - e^{-t}}{ e^t + e^{-t}})dt\\ =4\int(\frac{ e^t - e^{-t}}{ (e^t + e^{-t})^2})dt\\

Put (et+et)=u(e^t + e^{-t})=u

(etet)dt=du\therefore (e^t-e^{-t})dt=du\\

So, I=4(1u2)duI=4\int (\frac{1}{u^2})du\\

=4u+c=-\frac{4}{u}+c where cc is the constant of integration.

=4(et+et)+c=4(ex+ex)+c=-\frac{4}{(e^t + e^{-t})}+c\\ =-\frac{4}{(e^{\sqrt{x}} + e^{-\sqrt{x}})}+c\\


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment