Question

\[\int x^3 \cos x^2 dx\]

Answer 1

\[\int x^3 \cos x^2 \text{ dx }\]
\[\text{  Let x}^2 = t \]
\[ \Rightarrow 2x = \frac{dt}{dx}\]
\[ \Rightarrow dx = \frac{dt}{2x}\]
\[ = \frac{1}{2}\left[ \int t \text{ cos t dt } \right]\]
\[\text{Taking t as the first function and cos t as the second function} . \]
\[ = \frac{1}{2}\left[ t\sin t - \int\text{ sin t dt } \right]\]
\[ = \frac{1}{2}\left[ t\sin t + \cos t \right] . . . (1) \]
\[\text{Substituting the value of t in eq} \text{  (1) } \]
\[ = \frac{x^2 \sin x^2}{2} + \frac{\cos x^2}{2} + c\]