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Chapter 5: Q. 10 (page 428)

Find three integrals in Exercises 27–70 for which either algebra or u-substitution is a better strategy than integration by parts.

Short Answer

Expert verified

$\int x \sin x^{2} d x$ , $\int 3 x e^{x^{2}} d x$ , $\int \frac{3 x + 1}{x} d x$

Step by step solution

01

Step 1. Given information

Integrals should be easily solved by u- substitution or algebra instead of integration by parts.

02

Step 2. Observing integrals

The integral $\int x \sin x^{2} d x$ can easily be solved by using u- substitution as for $u = x^{2}$ , the derivative du of u equals $2 x d x$ . Hence, the transformed integral $\frac{1}{2} \int \sin u d u$ can be solved simply.
The integral $\int 3 x e^{x^{2}} d x$ can easily be solved by using u- substitution as for $u = x^{2}$ , the derivative du of u equals $2 x d x$ . Hence, the transformed integral $3 \int \frac{e^{u}}{2} d u$ can be solved simply.
The integral $\int \frac{3 x + 1}{x} d x$ can easily be solved by using algebra because the integrand can be simplified as $3 + \frac{1}{x}$ . Hence, the transformed integral $\int 3 + \frac{1}{x} d x$ can be solved simply.

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Most popular questions from this chapter

Solve given definite integral.

$\int_{- 1}^{1} x^{3} \sqrt{9 x^{2} - 1} d x$

Show by differentiating (and then using algebra) that $- \cot (\sin^{- 1} x)$ and $- \frac{\sqrt{1 - x^{2}}}{x}$ are both antiderivatives of $\frac{1}{x^{2} \sqrt{1 - x^{2}}}$ . How can these two very different-looking functions be an antiderivative of the same function?

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

Solve the integral: $\int x \ln (x) d x$

Solve the integral

$\int \sqrt{3 - x} \sqrt{x - 1} d x$

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