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Chapter 5: Q. 42 (page 417)

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$鈭� x \ln (e^{x^{2} + 1}) d x$

Short Answer

Expert verified

The solution of the given integral is $鈭� x \ln (e^{x^{2} + 1}) d x = \frac{1}{2} e^{\ln (e^{x^{2} + 1})} + C$ .

Step by step solution

01

Step 1. Given Information

Solving the given integrals.

$鈭� x \ln (e^{x^{2} + 1}) d x$

02

Step 2. Solving the given integral using substitution method.

Let

$u = \ln (e^{x^{2} + 1}) \frac{d u}{d x} = \frac{2 x}{e^{x^{2} + 1}} d u = \frac{2 x}{e^{x^{2} + 1}} d x \frac{e^{x^{2} + 1}}{2} d u = x d x$

03

Step 3. In this substitution after differentiation not at all in the form of f'(u(x))u'(x).

A clever change of variables will allow us to rewrite the integral so that it can be algebraically simplified.

$u = \ln (e^{x^{2} + 1}) e^{u} = e^{x^{2} + 1}$

$\frac{e^{u}}{2} d u = x d x$

04

Step 4. This substitution changes the integral into

$鈭� x \ln (e^{x^{2} + 1}) d x = \frac{1}{2} 鈭� e^{u} d u 鈭� x \ln (e^{x^{2} + 1}) d x = \frac{1}{2} e^{u} + C 鈭� x \ln (e^{x^{2} + 1}) d x = \frac{1}{2} e^{\ln (e^{x^{2} + 1})} + C$

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Most popular questions from this chapter

Why don鈥檛 we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution $x = a \tan 鈦� u$ when we see $x^{2} + a^{2}$ , even though we can鈥檛 use the substitution $x = a \sin 鈦� u$ unless the integrand involves the square root of $a^{2} 鈭� x^{2}$ ? (Hint: Think about domains.)

For each integral in Exercises 5鈥�8, write down three integrals that will have that form after a substitution of variables.

$鈭� \sin u d u$

Problem Zero: Read the section and make your own summary of the material.

Find three integrals in Exercises 27鈥�70 for which a good strategy is to use integration by parts with $u = x$ and dv the remaining part.

Consider the integral $鈭� \sin x \cos x d x$ .

(a) Solve this integral by using u-substitution with $u = \sin x$ and $d u = \cos x d x$ .

(b) Solve the integral another way, using u-substitution with $u = \cos x$ and $d u = 鈭� \sin x d x$ .

(c) How must your two answers be related? Use algebra to prove this relationship.

See all solutions

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