\( \int 3x^{2}e^{x^{3}}\,dx \) is equal to : (where \( C \) is an arbitrary constant)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( e^{x^{3}}+C \)
The factor \( 3x^{2} \) is the derivative of \( x^{3} \), so the substitution \( t=x^{3} \), \( dt=3x^{2}dx \) gives
\( \displaystyle \int 3x^{2}e^{x^{3}}\,dx=\int e^{t}\,dt \)
\( \displaystyle =e^{t}+C \)
\( \displaystyle =e^{x^{3}}+C \)