Solve these integrals

5)∫cot​−1​​e​x​​⋅e​−x​​dx

−∫tan​−1​​e​−x​​de​−x​​

Applying integration by parts,

−e​−x​​tan​−1​​(e​−x​​)+​2​​1​​ln∣e​−2x​​+1∣+C

​2​​1​​ln∣e​2x​​+1∣−x−cot​−1​​(e​x​​)e​−x​​+C

Ans-C

∫​0​√​lnπ/2​​​​​cos(e​x​2​​​​)2xe​x​​​2​​dx=∫​1​π/2​​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)

1c and 5 c i think

Sry sir 3 b