Q. 35 Solve 鈭�1x2+9dx聽 the following... [FREE SOLUTION]

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Chapter 5: Q. 35 (page 464)

Solve $鈭� \frac{1}{x^{2} + 9} d x$ the following two ways:
(a) with the trigonometric substitution x = 3 tan u;
(b) with algebra and the derivative of the arctangent.

Short Answer

Expert verified

Part (a) The solution of the integral is $\frac{1}{3} \tan^{- 1} \frac{x}{3} + C .$

Part (b) The solution of the integral is $\frac{1}{3} \tan^{- 1} \frac{x}{3} + C .$

Step by step solution

Part (a) Step 1. Given Information.

The given integral is $鈭� \frac{1}{x^{2} + 9} d x .$

Part (a) Step 2. Solve.

We have to solve the integral with the substitution $x = 3 \tan u,$ so the derivative is $d x = 3 s e c^{2} u d u .$

Let's solve the integral by substituting x,

$鈭� \frac{1}{x^{2} + 9} d x = 鈭� \frac{1}{{(3 \tan u)}^{2} + 9} (3 s e c^{2} u) d u = 鈭� \frac{3 s e c^{2} u}{9 \tan^{2} u + 9} d u = 鈭� \frac{3 s e c^{2} u}{9 (\tan^{2} u + 1)} d u = \frac{1}{3} 鈭� \frac{s e c^{2} u}{\tan^{2} u + 1} d u Usethe identity 1 + t a n^{2} x = s e c^{2} x = \frac{1}{3} 鈭� \frac{s e c^{2} u}{s e c^{2} u} d u = \frac{1}{3} 鈭� 1 d u = \frac{1}{3} u + C Substitute back u, = \frac{1}{3} \tan^{- 1} (\frac{x}{3}) + C$

Part (b) Step 1. Solve.

We have to solve the integral with algebra and the derivative of the arctangent.

As we know the derivative of the arctangent is $\frac{1}{x^{2} + 1} .$

Now, let's solve the integral,

$鈭� \frac{1}{x^{2} + 9} d x = 鈭� \frac{1}{9 (\frac{x^{2}}{9} + 1)} d x = \frac{1}{9} 鈭� \frac{1}{({(\frac{x}{3})}^{2} + 1)} d x Use the derivative of arctangent = \frac{1}{9} 脳 3 \tan^{- 1} \frac{x}{3} + C = \frac{1}{3} \tan^{- 1} \frac{x}{3} + C$

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91影视

Solve $鈭� \frac{1}{x^{2} + 9} d x$ the following two ways:
(a) with the trigonometric substitution x = 3 tan u;
(b) with algebra and the derivative of the arctangent.

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Solve.

Part (b) Step 1. Solve.

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91影视

Solve 鈭�1x2+9dx the following two ways:(a) with the trigonometric substitution x = 3 tan u;(b) with algebra and the derivative of the arctangent.

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Solve.

Part (b) Step 1. Solve.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Solve $鈭� \frac{1}{x^{2} + 9} d x$ the following two ways:
(a) with the trigonometric substitution x = 3 tan u;
(b) with algebra and the derivative of the arctangent.