Question

Find the integral of \(\frac{e^{-x} (1-x)}{sin^2⁡(xe^{-x})}\).

(a) cot⁡xe^-x+C

(b) -cot⁡xe^-x+C

(c) -cot⁡xe^x+C

(d) -cos^2⁡xe^-x+C

I have been asked this question by my college professor while I was bunking the class.

My query is from Methods of Integration-2 topic in section Integrals of Mathematics – Class 12

Answer 1

Right choice is (b) -cot⁡xe^-x+C

To explain: \(\int \frac{e^{-x} (1-x)}{sin^2⁡(xe^{-x})} dx\)

Let xe^-x=t

Differentiating w.r.t x, we get

\(-xe^{-x}+e^{-x} dx=dt\)

e^-x (1-x)dx=dt

\(\int \frac{e^{-x} (1-x)}{sin^2⁡(xe^{-x})} dx=\int \frac{dt}{sin^2⁡t}\)

=\(\int cosec^2 \,t \,dt\)

=-cot⁡t+C

Replacing t with xe^-x, we get

\(\int \frac{e^{-x} (1-x)}{sin^2⁡(xe^{-x})} dx=-cot⁡xe^{-x}+C\).