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Question:

Given ∫ex(tanx+1)secxdx=exf(x)+c. Find f(x).

Solution:

It is given that
I=∫ex(tanx+1)secxdx=exf(x)+C (1)
Consider
I=∫ex(tanx+1)secxdx=∫ex(secxtanx+secx)dx=∫exsecxdx+∫ex(secxtanx)dx
=secx∫exdx−∫[d(secx)/dx∫exdx]dx+∫ex(secxtanx)dx+C
=exsecx+C (2)
Comparing equations (1) and (2) we get
f(x)=secx.