Q. 58 Solve each of the integrals. Som... [FREE SOLUTION]

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Chapter 5: Q. 58 (page 465)

Solve each of the integrals. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \sqrt{x^{2} + 6 x + 18} d x$ .

Short Answer

Expert verified

The value is,

$\frac{9}{2} (\sin h^{- 1} (\frac{x + 3}{3})) + \frac{9}{4} \sin h (2 \sin h^{- 1} (\frac{x + 3}{3})) + C$ .

Step by step solution

Step 1. Given Information.

The integral is,

$鈭� \sqrt{x^{2} + 6 x + 18} d x$ .

Step 2. Simplifying the integral.

$鈭� \sqrt{x^{2} + 6 x + 18} d x = 鈭� \sqrt{{(x + 3)}^{2} + 9} d x = 鈭� \sqrt{u^{2} + 9} d u [u = x + 3, d u = d x]$

To solve the integral, let $u = 3 \sin h (v)$

Now, $u = 3 \sin h (v) d u = 3 \cos h (v) d v$ .

Using the identity, $\sin h^{2} (v) + 1 = \cos h^{2} (v)$ we find the values,

$\sqrt{u^{2} + 9} d u = \sqrt{9 \sin h^{2} (v) + 9} = 3 \sqrt{\cos h^{2} (v)}$

Step 3. Solving the integral.

The integral solution is,

$鈭� \sqrt{u^{2} + 9} d u = 鈭� 9 \cos h^{2} (v) d v = 9 鈭� \frac{1}{2} d v + 9 鈭� \cos h^{2} (2 v) d v = \frac{9}{2} v + \frac{9}{2} 鈭� \frac{1}{2} \cos h (w) d w [w = 2 v, d w = 2 d v] = \frac{9}{2} v + \frac{9}{4} \sin h (w) = \frac{9}{2} v + \frac{9}{4} \sin h (2 v) = \frac{9}{2} (\sin h^{- 1} (\frac{u}{3})) + \frac{9}{4} \sin h (2 \sin h^{- 1} (\frac{u}{3})) = \frac{9}{2} (\sin h^{- 1} (\frac{x + 3}{3})) + \frac{9}{4} \sin h (2 \sin h^{- 1} (\frac{x + 3}{3})) + C$

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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.
$鈭� \sqrt{x^{2} + 6 x + 18} d x$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Simplifying the integral.

Step 3. Solving the integral.

One App. One Place for Learning.

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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+6x+18dx.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Simplifying the integral.

Step 3. Solving the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Calculus

Applied Mathematics

Theoretical and Mathematical Physics

Statistics

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.
$鈭� \sqrt{x^{2} + 6 x + 18} d x$ .