1.

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+CI 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»

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

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


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