Chapter 5: Q. 12 (page 428)

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

Expert verified

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

Step by step solution

Integration by parts should be applied twice.

The integral $\int x^{2} \cos x d x$ is the product of two functions namely $x^{2}$ and $\cos x d x$ . It can be solved by taking $u = x^{2}$ , localid="1648731171575" $d v = \cos x d x$ . 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 $\int x^{2} e^{3 x} d x$ is the product of two functions namely $x^{2}$ and $e^{3 x} d x$ . It can be solved by taking $u = x^{2}$ , $d v = e^{3 x} d x$ . 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 $\int e^{2 x} \sin x d x$ is the product of two functions namely $e^{2 x}$ and $\sin x d x$ . It can be solved by taking $u = e^{2 x}$ , $d v = \sin x d x$ . 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.

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