Q. 56 Solve each of the integrals in E... [FREE SOLUTION]

Chapter 5: Q. 56 (page 418)

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$鈭� \frac{x^{2}}{\sqrt{x + 1}} d x$

Short Answer

Expert verified

The solution of the given integral is $鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = \frac{2}{5} {(x + 1)}^{5 / 2} - \frac{4}{3} {(x + 1)}^{3 / 2} + 2 {(x + 1)}^{1 / 2} + C$ .

Step by step solution

Step 1. Given Information

Solving the given integrals.

$鈭� \frac{x^{2}}{\sqrt{x + 1}} d x$

Step 2. Using the substitution method.

$u = x + 1 \frac{d u}{d x} = 1 d u = d x$

$x = u - 1$

Step 3. This substitution changes the integral into

$鈭� \frac{x^{2}}{\sqrt{x + 1}} d x = 鈭� \frac{{(u - 1)}^{2}}{\sqrt{u}} du 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = 鈭� \frac{{{{(u)}^{2} - 2 脳 1 脳 u + (1)}^{2}}}{u^{1 / 2}} du 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = 鈭� {(u}^{2} - 2 u + 1) u^{- 1 / 2} du 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = 鈭� {(u}^{2} 路 u^{- 1 / 2} - 2 u 路 u^{- 1 / 2} + 1 路 u^{- 1 / 2}) du 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = 鈭� (u^{2 - 1 / 2} - 2 u^{1 - 1 / 2} + u^{- 1 / 2}) du 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = 鈭� (u^{3 / 2} - 2 u^{1 / 2} + u^{- 1 / 2}) du$

Step 4. After simlifying.

$鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = 鈭� u^{3 / 2} du - 2 鈭� u^{1 / 2} du + 鈭� u^{- 1 / 2} du 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = [\frac{u^{3 / 2 + 1}}{3 / 2 + 1}] - 2 [\frac{u^{1 / 2 + 1}}{1 / 2 + 1}] + [\frac{u^{- 1 / 2 + 1}}{- 1 / 2 + 1}] + C 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = [\frac{u^{5 / 2}}{5 / 2}] - 2 [\frac{u^{3 / 2}}{3 / 2}] + [\frac{u^{1 / 2}}{1 / 2}] + C 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = \frac{2}{5} u^{5 / 2} - 2 路 \frac{2}{3} 路 u^{3 / 2} + 2 u^{1 / 2} + C 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = \frac{2}{5} u^{5 / 2} - \frac{4}{3} 路 u^{3 / 2} + 2 u^{1 / 2} + C 鈭� \frac{x^{2}}{\sqrt{x + 1}} dx = \frac{2}{5} {(x + 1)}^{5 / 2} - \frac{4}{3} {(x + 1)}^{3 / 2} + 2 {(x + 1)}^{1 / 2} + C$

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Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$鈭� \frac{x^{2}}{\sqrt{x + 1}} d x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Using the substitution method.

Step 3. This substitution changes the integral into

Step 4. After simlifying.

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91影视

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�x2x+1dx

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Using the substitution method.

Step 3. This substitution changes the integral into

Step 4. After simlifying.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Discrete Mathematics

Decision Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$鈭� \frac{x^{2}}{\sqrt{x + 1}} d x$