Q. 68 Solve each of the integrals in E... [FREE SOLUTION]

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Chapter 5: Q. 68 (page 465)

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.
$鈭� x^{5} \sqrt{x^{2} - 1} d x$

Short Answer

Expert verified

The value is $(\frac{2 x^{2}}{5} + \frac{1}{7} {(x^{2} - 1)}^{2} - \frac{1}{15}) {(x^{2} - 1)}^{\frac{3}{2}} + C$

Step by step solution

Step 1. Given information

An expression is given as $鈭� x^{5} \sqrt{x^{2} - 1} d x$

Step 2. Solving integral

Let

$u = x^{2} d u = 2 x d x$

Now substitute the value and simplify as,

$鈭� x^{5} \sqrt{x^{2} - 1} d x = 鈭� \frac{u^{2}}{2} \sqrt{u - 1} d u = \frac{1}{2} 鈭� u^{2} \sqrt{u - 1} d u = \frac{1}{2} 鈭� 2 v^{2} {(v^{2} + 1)}^{2} d v = 鈭� (v^{6} + 2 v^{4} + v^{2}) d v = 鈭� v^{6} d v + 2 鈭� v^{4} d v + 鈭� v^{2} d v = \frac{v^{7}}{7} + \frac{2 v^{5}}{5} + \frac{v^{3}}{3} + C = \frac{(\sqrt{u - 1})^{7}}{7} + \frac{2 (\sqrt{u - 1})^{5}}{5} + \frac{(\sqrt{u - 1})^{3}}{3} + C = \frac{1}{7} {(x^{2} - 1)}^{\frac{7}{2}} + \frac{2}{5} {(x^{2} - 1)}^{\frac{5}{2}} - \frac{1}{3} {(x^{2} - 1)}^{\frac{3}{2}} + C = (\frac{2 x^{2}}{5} + \frac{1}{7} {(x^{2} - 1)}^{2} - \frac{1}{15}) {(x^{2} - 1)}^{\frac{3}{2}} + C$

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91影视

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.
$鈭� x^{5} \sqrt{x^{2} - 1} d x$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solving integral

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91影视

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�x5x2-1dx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solving integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Decision Maths

Mechanics Maths

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

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.
$鈭� x^{5} \sqrt{x^{2} - 1} d x$