Q. 12 - Calculus

Question

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

Step-by-Step Solution

Answer

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

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 $u=x^2$, $dv=\cos xdx$. 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.