Apply Integration By Parts Twice
Evaluate
To solve the integral of a product like this, involving a polynomial and a trigonometric function, the key method to apply is integration by parts. In cases like this, we often need to apply the technique twice to fully simplify the expression.
First, as part of the strategy, you’ll choose which function to differentiate and which to integrate. Since differentiating the polynomial term, x^2, reduces its degree, it’s a good candidate for differentiation. The trigonometric function, cos(x), is easier to integrate, so it will remain as is during the first step.
After applying integration by parts once, you will be left with a simpler integral. However, this new integral will still involve a polynomial term (now linear instead of quadratic) and a trigonometric function. This is where you apply integration by parts again to complete the simplification process.
This technique of repeated integration by parts is effective for handling products of polynomials and trigonometric functions, especially when one term reduces in complexity with each application of the rule. After completing the second integration by parts, you’ll arrive at the solution to the original integral.