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Chapter 5: Q. 76 (page 452)

Solve each of the definite integrals in Exercises 67–76.
$\int_{0}^{π / 4} \frac{\sin x}{\cos^{2} x} d x$ .

Short Answer

Expert verified

The answer is $- 1 + \sqrt{2}$ .

Step by step solution

01

Step 1. Given information

The integral is $\int_{0}^{π / 4} \frac{\sin x}{\cos^{2} x} d x$ .

02

Step 2. Explanation.

To solve the integral use substitution method.

Substitute $u = \cos x$ so

$d u = - \sin x d x - d u = \sin x d x$

So the integral becomes,

$\int_{1}^{\frac{1}{\sqrt{2}}} - \frac{1}{u^{2}} d u$

Solve the integral as follows,

$\int_{1}^{\frac{1}{\sqrt{2}}} - \frac{1}{u^{2}} d u = - \int_{\frac{1}{\sqrt{2}}}^{1} - \frac{1}{u^{2}} d u = \int_{\frac{1}{\sqrt{2}}}^{1} \frac{1}{u^{2}} d u = {[- \frac{1}{u}]}_{\frac{1}{\sqrt{2}}}^{1} = - 1 + \sqrt{2}$

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Most popular questions from this chapter

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$\int \frac{\sqrt{4 - x^{2}}}{x^{2}} d x$

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$\int x \sqrt{x^{2} + 1} d x$

Why don’t 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 $x = a \tan u$ when we see $x^{2} + a^{2}$ , even though we can’t use the substitution $x = a \sin u$ unless the integrand involves the square root of $a^{2} - x^{2}$ ? (Hint: Think about domains.)

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An integral with which we could reasonably apply trigonometric substitution with $x = \tan u$ .

(b) An integral with which we could reasonably apply trigonometric substitution with $x = 4 \sec u$ .

(c) An integral with which we could reasonably apply trigonometric substitution with $x - 2 = 3 \sin u$ .

Solve the integral: $\int x \sin {(x)}^{2} d x$ .

See all solutions

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