Final answer:
The integral of x² cos(2x) is evaluated using the method of integration by parts, where we select u=x², and dv=cos(2x)dx. This may require multiple applications of integration by parts to fully solve the integral.
Step-by-step explanation:
To evaluate the integral of x² cos(2x), we can use the method of integration by parts. This method is designed to handle products of functions, which is exactly what we have here. The integration by parts formula is ∫u v dx = uv - ∫ v du, where u and dv are chosen from the integrand.
First, we let u = x², which implies that du = 2x dx. Then, we take dv = cos(2x) dx, which results in v = ⅓sin(2x) after integrating.
Applying integration by parts, we get:
- ∫x² cos(2x) dx = x²(⅓sin(2x)) - ∫(⅓sin(2x))(2x dx)
- The resulting integral can be simplified and may require integration by parts to be applied again or other integration methods.
It's important to note that we might need to perform integration by parts more than once to completely evaluate the integral. Additionally, it is worth mentioning that, for integrals involving trigonometric functions like sine and cosine, which are periodic, the average over a complete cycle would be the same for both sine squared and cosine squared terms.