Question

\[\int x^3 \sin \left( x^4 + 1 \right) dx\]

Answer 1

\[\int x^3 \cdot \sin \left( x^4 + 1 \right) dx\]
\[\text{Let x}^4 + 1 = t\]
\[ \Rightarrow 4 x^3 = \frac{dt}{dx}\]
\[ \Rightarrow x^3 \text{dx} = \frac{dt}{4}\]
\[Now, \int x^3 \cdot \sin \left( x^4 + 1 \right) dx\]
\[ = \frac{1}{4}\int\sin \left( t \right) dt\]
\[ = \frac{1}{4}\left[ - \cos t \right] + C\]
\[ = \frac{1}{4}\left[ - \cos \left( x^4 + 1 \right) \right] + C\]