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Chapter 5: Q. 79 (page 465)

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

Short Answer

Expert verified

$\frac{40 \sqrt{5}}{3} - \frac{8}{3} .$

Step by step solution

01

Step1. Given Information

The integral is as follows.

$\int_{0}^{4} x \sqrt{x^{2} + 4} d x$

The objective is to solve the integral.

02

Step2. Assumption

The integral is solved below.

$\int x \sqrt{x^{2} + 4} d x$

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

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

03

Step3. Solution

The definite integral is solved below.

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

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

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Most popular questions from this chapter

Solve the integral: $\int x \cos (x) d x$ .

Consider the integral $\int \frac{1}{x^{2} \sqrt{1 - x^{2}}} d x$ from the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution $u = \sin^{- 1} x$ to solve this integral.

(b) Use the trigonometric substitution $x = \sin u$ to solve the integral.

(c) Compare and contrast the two methods used in parts (a) and (b).

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: $f (x) = \frac{x + 1}{x - 1}$ is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of $\frac{x^{2} + 1}{x^{2} (x - 3)}$ is of the form $\frac{A}{x^{2}} + \frac{B}{x - 3}$

(g) True or False: The partial-fraction decomposition of $\frac{x^{2} + 1}{x^{2} (x - 3)}$ is of the form $\frac{B x + C}{x^{2}} + \frac{A}{x - 3}$ .

(h) True or False: Every quadratic function can be written in the form $A {(x - k)}^{2} + C$

Suppose v(x) is a function of x. Explain why the integral

of dv is equal to v (up to a constant).

Find three integrals in Exercises 39–74 that can be solved by using a trigonometric substitution of the form $x = a \tan u$ .

See all solutions

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