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Expert-verified Found in: Page 428 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Find three integrals in Exercises 27–70 for which a good strategy is to apply integration by parts twice.

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

See the step by step solution

## Step 1. Given information

Integration by parts should be applied twice.

## Step 2. Observing the integrals

• The integral $\int {x}^{2}\mathrm{cos}xdx$ is the product of two functions namely ${x}^{2}$ and $\mathrm{cos}xdx$. It can be solved by taking $u={x}^{2}$, localid="1648731171575" $dv=\mathrm{cos}xdx$. 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 $\int {e}^{2x}\mathrm{sin}xdx$ is the product of two functions namely ${e}^{2x}$ and $\mathrm{sin}xdx$. It can be solved by taking $u={e}^{2x}$, $dv=\mathrm{sin}xdx$. 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. ### Want to see more solutions like these? 