Chapter 5: Q. 79 (page 465)

Solve given definite integrals.
${鈭�}_{0}^{4} x \sqrt{x^{2} + 4} d x$

Short Answer

Expert verified

$\frac{40 \sqrt{5}}{3} 鈭� \frac{8}{3} .$

Step by step solution

Step1. Given Information

The integral is as follows.

${鈭�}_{0}^{4} 鈥� x \sqrt{x^{2} + 4} d x$

The objective is to solve the integral.

Step2. Assumption

The integral is solved below.

$鈭� x \sqrt{x^{2} + 4} d x$

$= 鈭� \frac{\sqrt{u}}{2} d u [u = x^{2} + 4, d u = 2 x d x]$

$\begin{aligned} = \frac{1}{2} 鈭� \sqrt{u} d u \\ = \frac{u^{\frac{3}{2}}}{3} \\ = \frac{{(x^{2} + 4)}^{\frac{3}{2}}}{3} \\ = \frac{1}{3} {(x^{2} + 4)}^{\frac{3}{2}} \end{aligned}$

Step3. Solution

The definite integral is solved below.

$\begin{aligned} {鈭�}_{0}^{4} 鈥� x \sqrt{x^{2} + 4} d x \\ = {[\frac{1}{3} {(x^{2} + 4)}^{\frac{3}{2}}]}_{0}^{4} \\ = [\frac{1}{3} {(4^{2} + 4)}^{\frac{3}{2}}] 鈭� [\frac{1}{3} {(0^{2} + 4)}^{\frac{3}{2}}] \\ = \frac{40 \sqrt{5}}{3} 鈭� \frac{8}{3} \end{aligned}$

Therefore, the value is $\frac{40 \sqrt{5}}{3} 鈭� \frac{8}{3} .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve given definite integrals.
${鈭�}_{0}^{4} x \sqrt{x^{2} + 4} d x$

Short Answer

Step by step solution

Step1. Given Information

Step2. Assumption

Step3. Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Statistics

Calculus

Mechanics Maths

Geometry

Study anywhere. Anytime. Across all devices.

91影视

Solve given definite integrals.鈭�04xx2+4dx

Short Answer

Step by step solution

Step1. Given Information

Step2. Assumption

Step3. Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Statistics

Calculus

Mechanics Maths

Geometry

Study anywhere. Anytime. Across all devices.

Solve given definite integrals.
${鈭�}_{0}^{4} x \sqrt{x^{2} + 4} d x$