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

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

• A. x^2/2 + C
• B. -x cos(x) + sin(x) + C
• C. cos(x) + C
• D. -x^2 sin(x) + C

Answer: (B)

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.

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.