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Chapter 5: Q 17. (page 495)

Fill in the blanks to complete each of the following integration formulas.
$鈭� \csc x d x =____$

Short Answer

Expert verified

The result is $\ln |\tan (\frac{x}{2})| + C$ .

Step by step solution

01

Step 1. Given information.

Consider the given integration,

$鈭� \csc x d x$

02

Step 2. Simplify.

Simplify the given integration expression.

$\begin{array}{rcl} 鈭� \csc x d x & = & 鈭� \frac{\csc x (\csc x - \cot x)}{\csc x - \cot x} d x \\ = & 鈭� \frac{\csc^{2} x - \csc x \cot x}{\csc x - \cot x} d x \end{array}$

03

Step 3. Find the integration.

Apply the integration method.

$\begin{array}{rcl} 鈭� \csc x d x & = & 鈭� \frac{\csc^{2} x - \csc x \cot x}{\csc x - \cot x} d x [t = \csc x - \cot x, d t = - \cot (x) \csc (x) + \csc^{2} (x)] \\ = & 鈭� \frac{1}{t} d t \\ = & \ln |t| + C \\ = & \ln |\csc x - \cot x| + C \\ = & \ln |\frac{1}{\sin x} - \frac{\cos x}{\sin x}| + C \\ = & \ln |\frac{1 - \cos x}{\sin x}| + C \\ = & \ln |\tan (\frac{x}{2})| + C \end{array}$

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Most popular questions from this chapter

Suppose $u (x) = x^{2}$ . Calculate and compare the values of the following definite integrals:

role="math" localid="1648786835678" ${鈭�}_{- 1}^{5} u^{2} d u, {鈭�}_{x = - 1}^{x = 5} u^{2} d u and {鈭�}_{u (- 1)}^{u (5)} u^{2} d u$

Explain why using trigonometric substitution with $x = a \tan u$ often involves a triangle with side lengths a and x and hypotenuse of length $\sqrt{x^{2} + a^{2}}$

Solve $鈭� \frac{x + 2}{{(x^{2} + 4 x)}^{\frac{3}{2}}} d x$ the following two ways:

(a) with the substitution $u = x^{2} + 4 x;$

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.

Solve given definite integral.

${鈭�}_{1}^{2} 鈥� \frac{3}{{(9 鈭� 2 x^{2})}^{3 / 2}} d x$

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

$鈭� u^{2} d u$

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