Question

Find three integrals in Exercises 27–70 for which a good strategy is to apply integration by parts twice.

Short answer

$\int x^2\cos xdx$,$\int x^2{e}^{3x}dx$,$\int e^{2x}\sin xdx$

Step-by-step solution

Step 1. Given information

Integration by parts should be applied twice.

Step 2. Observing the integrals

• The integral $\int x^2\cos xdx$is the product of two functions namely $x^2$and $\cos xdx$. It can be solved by taking $u=x^2$, localid="1648731171575" $dv=\cos xdx$. However, even by using these substitutions, the integral is not solved properly which leads to the further integration of the obtained expression to obtain the simplified resultant.
• The integral $\int x^2{e}^{3x}dx$is the product of two functions namely $x^2$and $e^{3x}dx$. It can be solved by taking $u=x^2$, $dv=e^{3x}dx$. However, even by using these substitutions, the integral is not solved properly which leads to the further integration of the obtained expression to obtain the simplified resultant.
• The integral $\int e^{2x}\sin xdx$is the product of two functions namely $e^{2x}$and $\sin xdx$. It can be solved by taking $u=e^{2x}$, $dv=\sin xdx$. However, even by using these substitutions, the integral is not solved properly which leads to the further integration of the obtained expression to obtain the simplified resultant.