Question

\[\int x^2 e^{x^3} \cos x^3 dx\]

Answer 1

\[\text{Given integral is}, \]
\[\int x^2 e^{x^3} \cos \left( x^3 \right) \text{ dx }\]
\[\text{ Let x}^3 = t\]
\[ \Rightarrow 3 x^2 dx = dt\]
\[ \Rightarrow x^2 dx = \frac{dt}{3}\]
\[\text{Integral becomes}, \]
\[\frac{1}{3}\int e^t \text{ cos  t  dt}\]
\[ = \frac{1}{3}I . . . . . \left( 1 \right)\]
\[Where, I = \int e^t \text{ cos  t  dt }\]
\[I = \int e^t \text{ cos  t  dt}\]
`\text{Considering cos   t  as first function and` `\text{ e}^{t}`   ` \text{ as second function} `
\[I = \cos t e^t - \int - \text{ sin  t  e}^t dt\]
\[ \Rightarrow I = e^t \cos t + \int \text{ sin  t  e}^t dt\]
`\text{   Again considering  sin  t  as first function and` `\text{ e}^{t}`   ` \text{ as second function} `
\[I = e^t \cos t + \sin t e^t - \int \text{ cos  t  e}^t dt\]
\[ \Rightarrow I = e^t \cos t + \text{ sin  t e}^t - I\]
\[ \Rightarrow 2I = e^t \left( \text{ sin  t} + \cos t \right)\]
\[ \Rightarrow I = \frac{e^t}{2}\left( \sin t + \cos t \right)\]
\[ \therefore \int x^2\text{  e}^{x^3} \text{ cos} \left( x^3 \right) dx = \frac{1}{3}\left[ \frac{e^t}{2}\left( \sin t + \cos t \right) \right] + C \left[ \text{ From} \left( 1 \right) \right]\]
\[ = \frac{e^{x^3}}{6}\left( \sin x^3 + \cos x^3 \right) + C\]