Question

For each of the following integrals, indicate whether integration by substitution, integration by parts, or neither method is more appropriate. Do not evaluate the integrals.

∫ x sin x dx

i. substitution
ii. neither
iii. integration by parts

Answer 1

The integral of x sin x dx is best handled by using iii) integration by parts, which is designed for integrals involving the product of two functions.

Step-by-step explanation:

The appropriate method to evaluate the integral ∫ x sin x dx is integration by parts. Integration by substitution is typically used when there is a function and its derivative present in the integral, but in this case, x and sin(x) do not have that relationship. Instead, integration by parts is a more suitable choice as it is designed to handle the product of two functions, which is exactly what this integral is composed of.

In summary, for the integral ∫ x sin x dx, you would label x as one function (often u) and sin(x) as another (often dv), then apply the integration by parts formula u dv = uv - ∫ v du to solve.