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Chapter 5: Q. 35 (page 417)

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$鈭� \cot^{5} x \csc^{2} x d x$

Short Answer

Expert verified

The solution of the given integral is $鈭� \cot^{5} {xcsc}^{2} xdx = - \frac{1}{6} \cot^{6} x + C$ .

Step by step solution

01

Step 1. Given Information

Solving the given integrals.

$鈭� \cot^{5} x \csc^{2} x d x$

02

Step 2. Solving the given integral using substitution method.

Let

$u = \cot x \frac{d u}{d x} = - \csc^{2} x d u = - \csc^{2} x d u - d u = \csc^{2} x d u$

03

Step 3. This substitution changes the integral into

$鈭� \cot^{5} x \csc^{2} x d x = - 鈭� {(u)}^{5} du 鈭� \cot^{5} {xcsc}^{2} xdx = - [\frac{u^{5 + 1}}{5 + 1}] + C 鈭� \cot^{5} {xcsc}^{2} xdx = - \frac{u^{6}}{6} + C 鈭� \cot^{5} {xcsc}^{2} xdx = - \frac{1}{6} u^{6} + C 鈭� \cot^{5} {xcsc}^{2} xdx = - \frac{1}{6} \cot^{6} x + C$

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Most popular questions from this chapter

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

$鈭� x \sqrt{x^{2} + 1} d x$

Solve given definite integral.

${鈭�}_{1 / 4}^{1 / 2} 鈥� \frac{1}{x^{2} \sqrt{1 鈭� x^{2}}} d x$

Solve given definite integral.

${鈭�}_{1}^{2} 鈥� \frac{1}{x \sqrt{9 鈭� x^{2}}} d x$

Why don鈥檛 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 $x = a \tan 鈦� u$ when we see $x^{2} + a^{2}$ , even though we can鈥檛 use the substitution $x = a \sin 鈦� u$ unless the integrand involves the square root of $a^{2} 鈭� x^{2}$ ? (Hint: Think about domains.)

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

$鈭� \frac{1}{{(x^{2} + 4)}^{\frac{3}{2}}} d x$

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