Let I=∫(xsech(x).tanh(x))dx
Put x=t
∴2x1dx=dt⇒x1dx=2.dt
So, I=2∫(sech(t).tanh(t))dt
=2∫(et+e−t2)(et+e−tet−e−t)dt=4∫((et+e−t)2et−e−t)dt
Put (et+e−t)=u
∴(et−e−t)dt=du
So, I=4∫(u21)du
=−u4+c where c is the constant of integration.
=−(et+e−t)4+c=−(ex+e−x)4+c
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