Question

\[\int x^2 e^{x^3} \cos \left( e^{x^3} \right) dx\]

Answer 1

\[\int x^2 \cdot e^{x^3} \cdot \cos \left( e^{x^3} \right) dx\]
\[\text{Let e}^{x^3} = t\]
\[ \Rightarrow e^{x^3} \cdot 3 x^2 dx = dt\]
\[ \Rightarrow e^{x^3} \cdot x^2 dx = \frac{dt}{3}\]
\[Now, \int x^2 \cdot e^{x^3} \cdot \cos \left( e^{x^3} \right) dx\]
\[ = \frac{1}{3}\int\cos\left( t \right) dt\]
\[ = \frac{1}{3}\left[ \sin t \right] + C\]
\[ = \frac{1}{3}\left[ \sin \left( e^{x^3} \right) \right] + C\]