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Chapter 12: Problem 3

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{x}{\sqrt{2+3 x}} d x, \text { Formula } 19 $$

Short Answer

Expert verified
Therefore, the indefinite integral \( \int \frac{x}{\sqrt{2+3x}} dx \) is \( \frac{2}{27}(2 + 3x)^\frac{3}{2} - \frac{4}{9}(2 + 3x)^\frac{1}{2} + C \)

Step by step solution

01

Identify the Problem

The problem lies in integrating the given function: \( \int \frac{x}{\sqrt{2+3x}} dx \). We need to identify a suitable substitution that will simplify the integral and make it match with Formula 19.
02

Apply Substitution

For the integral, let's substitute \( u = 2 + 3x \). Hence, the differential \( du = 3 dx \) and \( dx = \frac{du}{3} \). Also notice x can be expressed as \( x = \frac{u-2}{3} \) from \( u = 2 + 3x \). Substitute these values back to the integral.
03

Simplify the Integral

Replacing these values in given integral, we obtain \( \int \frac{u-2}{3\sqrt{u}}. \frac{du}{3} = \frac{1}{9} \left( \int \frac{u}{\sqrt{u}} du - 2\int \frac{1}{\sqrt{u}} du \right) = \frac{1}{9} \left( \int u^{\frac{1}{2}}du -2 \int u^{-\frac{1}{2}} du \right) \)
04

Calculate the Integral

Now we can easily compute the integral which results in \( \frac{1}{9} ( \frac{2}{3} u^{\frac{3}{2}} - 2(2 u^{\frac{1}{2}}) )+ C = \frac{2}{27}u^{\frac{3}{2}} - \frac{4}{9}u^{\frac{1}{2}} + C \)
05

Back-Substitution

Do the back-substitution \( u = 2 + 3x \). The final result is \( \frac{2}{27}(2 + 3x)^\frac{3}{2} - \frac{4}{9}(2 + 3x)^\frac{1}{2} + C \)

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Most popular questions from this chapter

Consumer Trends The rate of change \(S\) in the number of subscribers to a newly introduced magazine is modeled by \(d S / d t=1000 t^{2} e^{-t}, 0 \leq t \leq 6\), where \(t\) is the time in years. Use Simpson's Rule with \(n=12\) to estimate the total increase in the number of subscribers during the first 6 years.

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{2} \frac{x}{\sqrt{4-x^{2}}} d x $$

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \int x^{2} e^{x} d x \quad \text { Integration by parts } $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} \frac{1}{x+1} d x $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{0}^{2} x^{3} d x, n=8 $$
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