Solve these integrals

5)\int cot^{-1}e^{x}\cdot e^{-x}dx

-\int \tan^{-1}{e^{-x}}de^{-x}

Applying integration by parts,

-e^{-x}\tan^{-1}(e^{-x})+\frac{1}{2}\ln|e^{-2x}+1|+C

\frac{1}{2}\ln|e^{2x}+1|-x-\cot^{-1}(e^{x})e^{-x}+C

Ans-C

\int^{\sqrt{\ln \pi/2}}_{0}\cos(e^{x^{2}})2xe^{x^{}2}dx=\int^{\pi/2}_{1}\cos(e^{x^{2}})de^{x^{2}}

1-\sin1

4) f(g(x))=x ...(1)
f'(g(x))g'(x)=1
g'(0)=1f'(g(0))
From(1)
f(g(0))=0
f(x) is zero when both upper & lower limit are equal.
f'(x)= 1√(1+x⁴)
f'(2)=1√17
Hence g'(0)=√17
(c)

3a