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Chapter 7: Problem 80

Evaluate the following integrals. Assume a and b are real numbers and \(n\) is an integer. $$\int \frac{x}{\sqrt{a x+b}} d x\left(\text { Use } u^{2}=a x+b\right.$$

Short Answer

Expert verified
Question: Evaluate the following integral using the given substitution: $$\int \frac{x}{\sqrt{a x+b}} dx$$ with the substitution \(u^2 = ax+b\). Answer: $$\int \frac{x}{\sqrt{a x+b}} dx = \frac{2}{a} (\sqrt{ax+b} + \frac{b}{\sqrt{ax+b}} + C)$$

Step by step solution

01

Perform the substitution

Given the substitution, \(u^2 = ax+b\). The task is to express the integral in terms of \(u\). To do this, we will both rewrite the original integral in terms of \(u\), and also rewrite \(dx\) in terms of \(du\).
02

Calculate the derivative of \(u\) with respect to \(x\)

Differentiate both sides of the equation \(u^2 = ax+b\) with respect to \(x\). Using the chain rule, we get: $$\frac{d(u^2)}{dx} = \frac{d(ax+b)}{dx}$$ This can be further simplified as: $$2u \frac{du}{dx} = a$$ Now we can solve for \(\frac{du}{dx}\): $$\frac{du}{dx} = \frac{a}{2u}$$ Then solve for \(dx\) in terms of \(du\): $$dx = \frac{2u}{a} du$$
03

Simplify the integral

Replace \(x\) in the original integral with \(u\) using \(u^2 = ax+b\). So \(x = \frac{u^2 - b}{a}\). The integral becomes: $$\int \frac{\frac{u^2 - b}{a}}{\sqrt{u^2}} \frac{2u}{a} du$$ Simplify the integral: $$\int \frac{2u(u^2 - b)}{au^2} du$$
04

Evaluate the indefinite integral

At this point, the remaining integral is quite simple to evaluate: $$\int \frac{2u(u^2 - b)}{au^2} du = \frac{2}{a} \int \frac{u(u^2 - b)}{u^2} du$$ $$= \frac{2}{a} \int \left(1 - \frac{b}{u^2}\right) du$$ Integrate this expression with respect to \(u\): $$= \frac{2}{a} (u - \frac{-b}{u} + C)$$
05

Write the final answer

Since we performed a substitution earlier, we need to express our answer in terms of \(x\) instead of \(u\). Replace \(u\) using the original substitution, \(u^2 = ax+b\): $$\int \frac{x}{\sqrt{a x+b}} dx = \frac{2}{a} (\sqrt{ax+b} + \frac{b}{\sqrt{ax+b}} + C)$$ This is the final result of the integral.

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