Question

Find three integrals in Exercises 27–70 for which a good strategy is to use integration by parts with $u=x$and dv the remaining part.

Short answer

$\int x{e}^{x}dx$,$\int x\sin xdx$,$\int x\cos xdx$

Step-by-step solution

Step 1. Given information

Integrals should be solved by using integration by parts and by taking$u=x$.

Step 2. Observing the integrals

• The integral $\int x{e}^{x}dx$is the product of two functions namely x and $e^x$. It can be solved easily by taking $u=x$, which makes dv equal to $e^{x}dx$.
• The integral $\int x\sin xdx$is the product of two functions namely x and $\sin x$. It can be solved easily by taking $u=x$, which makes dv equal to $\sin xdx$.
• The integral $\int x\cos xdx$is the product of two functions namely x and $\cos x$. It can be solved easily by taking $u=x$, which makes dv equal to $\cos xdx$.