Question

Integrate \(\frac{x^2}{e^{x^3}}\).

(a) –\(\frac{1}{(3e^{x^3})}+C\)

(b) \(\frac{1}{3e^{x^3}}+C\)

(c) –\(\frac{1}{e^{x^3}}+C\)

(d) e^x^3+C

The question was asked by my school principal while I was bunking the class.

This intriguing question originated from Methods of Integration-1 in division Integrals of Mathematics – Class 12

Answer 1

Right option is (a) –\(\frac{1}{(3e^{x^3})}+C\)

To explain: Let x^3=t

3x^2 dx=dt

x^2 dx=dt/3

∴\(\int \frac{x^2}{e^{x^3}} dx=\frac{1}{3} \int \frac{dt}{e^t}\)

=\(\frac{1}{3} \left (-e^{-t}\right )\)

Replacing t with x^3, we get

\(\int \frac{x^2}{e^{x^3}} dx=-\frac{1}{3e^{x^3}}+C\)