How do you compute the integral ∫ x sin(x) dx?

To compute the integral ∫ x sin(x) dx, we can use the method of integration by parts. The formula for integration by parts is given by:

∫ u dv = u v - ∫ v du

In this case, we can choose:

• u = x, which implies that du = dx

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

Now, applying the integration by parts formula:

∫ x sin(x) dx = u v - ∫ v du

= x (-cos(x)) - ∫ (-cos(x)) dx

= -x cos(x) + ∫ cos(x) dx

= -x cos(x) + sin(x) + C

The result simplifies to -x cos(x) + sin(x) + C, making the choice of B the correct answer.

This process highlights the importance of integration by parts as a technique to tackle products of functions, such as x and sin(x), in integrals. The other options don't match the derived result and are, therefore, not correct for this integral calculation.