Q. 63 Solve each of the integrals. Som... [FREE SOLUTION]

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Chapter 5: Q. 63 (page 465)

Solve each of the integrals. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \frac{\sqrt{x^{2} - 1}}{x} d x$ .

Short Answer

Expert verified

The value is,

$\sqrt{x - 1} \sqrt{x + 1} - \tan^{- 1} (\sqrt{x - 1} \sqrt{x + 1}) + C$ .

Step by step solution

Step 1. Given Information.

The integral is,

$鈭� \frac{\sqrt{x^{2} - 1}}{x} d x$ .

Step 2. Simplifying the function.

Let, $x = \cos h (u)$ .

Now, the derivation is,

$x = \cos h (u) d x = \sin h (u) d u$

Using the identity, $\cos h^{2} (u) - 1 = \sin h^{2} (u)$

So,

$\frac{\sqrt{x^{2} - 1}}{x} = \frac{\sqrt{\cos h^{2} (u) - 1}}{\cos h (u)} = \frac{\sqrt{\sin h^{2} (u)}}{\cos h (u)} = \frac{\sin h (u)}{\cos h (u)}$

Step 3. Solving the integral

The integral is,

$鈭� \frac{\sqrt{x^{2} - 1}}{x} d x = 鈭� \frac{\sin h^{2} (u)}{\cos h (u)} d u = 鈭� \frac{\sin h^{2} (u) \cos h (u)}{\sin h^{2} (u) + 1} d u = 鈭� \frac{v^{2}}{v^{2} + 1} d v [v = \sin h (u), d v = \cos h (u) d u] = 鈭� d v - 鈭� \frac{1}{v^{2} + 1} d v = v - \tan^{- 1} (v) + C = \sin h (u) - \tan^{- 1} (\sin h (u)) + C = \sin h (\cos h^{- 1} (x)) - \tan^{- 1} (\sin h (\cos h^{- 1} (x))) + C = \sqrt{x - 1} \sqrt{x + 1} - \tan^{- 1} (\sqrt{x - 1} \sqrt{x + 1}) + C$

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91影视

Solve each of the integrals. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \frac{\sqrt{x^{2} - 1}}{x} d x$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Simplifying the function.

Step 3. Solving the integral

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Solve each of the integrals. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.鈭�x2-1xdx.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Simplifying the function.

Step 3. Solving the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Solve each of the integrals. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \frac{\sqrt{x^{2} - 1}}{x} d x$ .