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Chapter 5: Q 77. (page 429)

Solve the definite integral.
$\int_{\frac{π}{4}}^{\frac{π}{2}} x c s c^{2} x d x$

Short Answer

Expert verified

The solution is $\frac{1}{4} (π + 2 \log 2)$ .

Step by step solution

01

Step 1. Given information.

The given integral is $\int_{\frac{π}{4}}^{\frac{π}{2}} x c s c^{2} x d x$ .

02

Step 2. First, solve the indefinite integral.

$\begin{array}{rcl} I & = & \int x c s c^{2} x d x \\ = & x \int c s c^{2} x d x - \int [\frac{d}{d x} (x) \int c s c^{2} x d x] d x \\ = & x (- c o t x) - \int 1 (- c o t x) d x \\ = & - x c o t x + \int c o t x d x \\ = & - x c o t x + \log |\sin x| \dots \dots \dots \dots \{\int c o t x d x = \log |\sin x|\} \end{array}$

03

Step 3. Now apply the limit of integration.

$\begin{array}{rcl} \int_{\frac{π}{4}}^{\frac{π}{2}} x c s c^{2} x d x & = & {[- x c o t x + \log |\sin x|]}_{\frac{π}{4}}^{\frac{π}{2}} \\ = & [- (\frac{π}{2}) c o t (\frac{π}{2}) + \log |\sin (\frac{π}{2})|] - [- (\frac{π}{4}) c o t (\frac{π}{4}) + \log |\sin (\frac{π}{4})|] \\ = & \frac{1}{4} (π + 2 \log 2) \end{array}$

04

Step 4. Simplified answer.

Hence, the required value is $\frac{1}{4} (π + 2 \log 2)$ .

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Most popular questions from this chapter

Solve given definite integral.

$\int_{4}^{5} \frac{1}{x \sqrt{x^{2} + 9}} d x$

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{x^{2} - 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving $\int \frac{1}{x^{2} + 4} d x .$

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $\int {(x^{2} + 4)}^{- 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} - a^{2}}$ .

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

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.

$\int \frac{1}{\sqrt{3 - x^{2}}} d x$

Solve the integral $\int x^{3} \sqrt{x^{2} - 1} d x$ three ways:

(a) with the substitution $u = x^{2} - 1,$ followed by back substitution;

(b) with integration by parts, choosing localid="1648814744993" $u = x^{2} and d v = x \sqrt{x^{2} - 1} d x;$

(c) with the trigonometric substitution x = sec u.

Explain how to use long division to write the improper fraction $\frac{172}{5}$ as the sum of an integer and a proper fraction.

See all solutions

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