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

Category: QuestionsIntegrate \frac{x^2}{e^{x^3}}.
Editor">Editor Staff asked 11 months ago

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
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

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