Chapter 5: Q. 42 (page 417)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

Solving the given integrals.

$\int x \ln (e^{x^{2} + 1}) d x$

Step 2. Solving the given integral using substitution method.

Let

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

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.

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

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

Step 4. This substitution changes the integral into

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

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the given integral using substitution method.

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

Step 4. This substitution changes the integral into

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Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)∫xln(ex2+1)dx

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the given integral using substitution method.

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

Step 4. This substitution changes the integral into

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Probability and Statistics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

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