Chapter 5: Q. 7 (page 417)

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.
$\int \frac{1}{\sqrt{u}} d u$

Short Answer

Expert verified

The three integrals $\int \frac{2 x + 1}{\sqrt{x^{2} + x}} d x, \int \frac{\sin x}{\sqrt{\cos x}} d x and \int \frac{1}{x \sqrt{\ln x}} d x$ will have form $\int \frac{1}{\sqrt{u}} d u$ after a substitution of variables.

Step by step solution

Step 1. Given Information

For given integral we have to write down three integrals that will have that form after a substitution of variables.
$\int \frac{1}{\sqrt{u}} d u$

Step 2. The first integrals that will have that form after a substitution of variables.

Let

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

This substitution changes the integral into

localid="1649155815093" $\int \frac{2 x + 1}{\sqrt{x^{2} + x}} d x$

Step 3. The first integrals that will have that form after a substitution of variables.

Let

$u = \cos x \frac{d u}{d x} = \sin x d u = \sin x d x$

This substitution changes the integral into

role="math" localid="1648747414306" $\int \frac{\sin x}{\sqrt{\cos x}} d x$

Step 4. The first integrals that will have that form after a substitution of variables.

Let

$u = \ln x \frac{d u}{d x} = \frac{1}{x} d u = \frac{1}{x} d x$

This substitution changes the integral into

$\int \frac{1}{x \sqrt{\ln x}} d x$

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For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.
$\int \frac{1}{\sqrt{u}} d u$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The first integrals that will have that form after a substitution of variables.

Step 3. The first integrals that will have that form after a substitution of variables.

Step 4. The first integrals that will have that form after a substitution of variables.

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For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.∫1udu

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The first integrals that will have that form after a substitution of variables.

Step 3. The first integrals that will have that form after a substitution of variables.

Step 4. The first integrals that will have that form after a substitution of variables.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Calculus

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.
$\int \frac{1}{\sqrt{u}} d u$