Question

What method should be used for the equation (either the substitution rule or the integration by parts). Next, evaluate the integrals given.

Answer 1

Given the integral on the picture, we can use integration by parts to find the antiderivative.

First, let u = t² an dv = cos3t dt. If we find the derivative of u and the integral of v, we get:

`\begin{gathered} u=t²\Rightarrow du=2tdt \\ dv=cos3tdt\Rightarrow v=(1)/(3)sin3t \end{gathered}`

then, using the formula for integration by parts, we have the following:

`\int t²cos3tdt=(1)/(3)t²sin3t-\int(2)/(3)tsin3tdt`

notice that the resulting integral on the right side also can be solved by parts. The solution of this integral is the following:

`\int(2)/(3)tsin3tdt=(2)/(9)tcos3t-(2)/(27)sin3t`

then, combining both results, we get:

`\int t²cos3tdt=(1)/(3)t²sin3t+(2)/(9)tcos3t-(2)/(27)sin3t+C`