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:

$\begin{gathered} u=t²\Rightarrow du=2tdt \\ dv=cos3tdt\Rightarrow v=\frac{1}{3}sin3t \end{gathered}$

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

$\int t²cos3tdt=\frac{1}{3}t²sin3t-\int\frac{2}{3}tsin3tdt$

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

$\int\frac{2}{3}tsin3tdt=\frac{2}{9}tcos3t-\frac{2}{27}sin3t$

then, combining both results, we get:

$\int t²cos3tdt=\frac{1}{3}t²sin3t+\frac{2}{9}tcos3t-\frac{2}{27}sin3t+C$