What is the result of ∫ x^2 sin(x) dx using integration by parts?

To evaluate the integral ∫ x² sin(x) dx using integration by parts, we apply the integration by parts formula:

∫ u dv = uv - ∫ v du

Here, we choose u and dv as follows:

• Let u = x², which gives us du = 2x dx.

• Let dv = sin(x) dx, leading to v = -cos(x).

Applying the integration by parts formula:

∫ x² sin(x) dx = -x² cos(x) - ∫ (-cos(x)) (2x) dx

= -x² cos(x) + 2 ∫ x cos(x) dx

Thus, we correctly arrive at the expression -x² cos(x) + 2 ∫ x cos(x) dx + C.

This confirms that the choice stating this result accurately reflects the integration by parts process performed on the original integral. Therefore, this option is the correct result for the integral ∫ x² sin(x) dx evaluated using integration by parts.