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Chapter 7: Problem 47

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int \frac{x}{\sqrt{2 x+3}} d x$$

Short Answer

Expert verified
Question: Evaluate the indefinite integral: $$\int \frac{x}{\sqrt{2 x+3}} d x$$ Answer: $$\int \frac{x}{\sqrt{2 x+3}} d x = \frac{1}{4}\left( \frac{2}{3}(2x+3)^{\frac{3}{2}} - 6(2x+3)^{\frac{1}{2}} \right) + D$$

Step by step solution

01

Substitution

Let's perform a substitution to simplify the integral. Let: $$u = 2x+3$$ Then differentiate with respect to x: $$\frac{d u}{d x} = 2$$ Now, we can express the differential dx in terms of du: $$d x = \frac{d u}{2}$$ Now, substitute the expressions above into the integral: $$\int \frac{x}{\sqrt{2 x+3}} d x$$ $$ = \int \frac{\frac{u-3}{2}}{\sqrt{u}} \cdot \frac{du}{2}$$
02

Simplify the expression

Now, let's simplify the expression in the integral: $$ = \frac{1}{4} \int \frac{u-3}{\sqrt{u}} du$$
03

Separate the terms

Separate the terms in the integral to deal with them individually: $$ = \frac{1}{4}\left( \int \frac{u}{\sqrt{u}} du - \int \frac{3}{\sqrt{u}} du \right)$$
04

Simplify the integrals

Simplify each integral to a more recognizable form: $$ = \frac{1}{4}\left( \int u^{\frac{1}{2}-1} du - 3 \int u^{-\frac{1}{2}} du \right)$$
05

Evaluate the integrals

Now, evaluate each integral using basic integration rule of power functions: $$\int x^n dx = \frac{x^{n+1}}{n+1}+C$$ $$ = \frac{1}{4}\left( \frac{u^{\frac{3}{2}}}{\frac{3}{2}} - 3 \frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C \right)$$
06

Simplify the expression

Simplify the expression further: $$ = \frac{1}{4}\left( \frac{2}{3}u^{\frac{3}{2}} - 6u^{\frac{1}{2}} + C \right)$$
07

Substitute back in terms of x

Substitute back in terms of x, by replacing u with 2x + 3: $$ = \frac{1}{4}\left( \frac{2}{3}(2x+3)^{\frac{3}{2}} - 6(2x+3)^{\frac{1}{2}} + C \right)$$
08

Combine constant terms

Combine the constant terms into a single constant term, let's say "D": $$ = \frac{1}{4}\left( \frac{2}{3}(2x+3)^{\frac{3}{2}} - 6(2x+3)^{\frac{1}{2}} \right) + D$$ So, the indefinite integral of the given function is: $$\int \frac{x}{\sqrt{2 x+3}} d x = \frac{1}{4}\left( \frac{2}{3}(2x+3)^{\frac{3}{2}} - 6(2x+3)^{\frac{1}{2}} \right) + D$$

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Most popular questions from this chapter

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$ F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t $$ where we assume that s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$ F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1} $$ Verify the following Laplace transforms, where a is a real number. $$f(t)=1 \longrightarrow F(s)=\frac{1}{s}$$

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{\sec \theta}{1+\sin \theta} d \theta$$

Evaluate the following integrals. Consider completing the square. $$\int \frac{d x}{\sqrt{(x-1)(3-x)}}$$

Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. $$\int \frac{2}{x\left(x^{2}+1\right)^{2}} d x$$

Let \(R\) be the region between the curves \(y=e^{-c x}\) and \(y=-e^{-c x}\) on the interval \([a, \infty),\) where \(a \geq 0\) and \(c \geq 0 .\) The center of mass of \(R\) is located at \((\bar{x}, 0)\) where \(\bar{x}=\frac{\int_{a}^{\infty} x e^{-c x} d x}{\int_{a}^{\infty} e^{-c x} d x} .\) (The profile of the Eiffel Tower is modeled by the two exponential curves.) a. For \(a=0\) and \(c=2,\) sketch the curves that define \(R\) and find the center of mass of \(R\). Indicate the location of the center of mass. b. With \(a=0\) and \(c=2,\) find equations of the lines tangent to the curves at the points corresponding to \(x=0.\) c. Show that the tangent lines intersect at the center of mass. d. Show that this same property holds for any \(a \geq 0\) and any \(c>0 ;\) that is, the tangent lines to the curves \(y=\pm e^{-c x}\) at \(x=a\) intersect at the center of mass of \(R\) (Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique \(332(2004): 571-584 .)\)
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