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Chapter 7: Problem 45

Evaluate the following integrals. $$\int \frac{d x}{x\left(x^{2}-1\right)^{3 / 2}}, x>1$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int \frac{dx}{x\left(x^2-1\right)^{3 / 2}}$$ Answer: $$-\frac{1}{2}\ln{|x|} + \frac{1}{2}\ln{|x-1|} + C$$

Step by step solution

01

Identify a trigonometric substitution that simplifies the integral

We will use the trigonometric substitution \(x = \cosh{t}\), since this substitution can help to simplify the given integral with \(x\left(x^{2}-1\right)^{3 / 2}\) in the denominator. Differentiating \(x\) with respect to \(t\) gives us: $$\frac{dx}{dt} = \sinh{t}$$ Which means, $$dx = \sinh{t} dt$$
02

Substitute the trigonometric expression and simplify

Replace \(x\) with \(\cosh{t}\) in the integral and \(dx\) with \(\sinh{t} dt\): $$\int \frac{d x}{x\left(x^{2}-1\right)^{3 / 2}} = \int \frac{\sinh{t} dt}{\cosh{t} \left(\cosh^{2}{t} - 1\right)^{3 / 2}}$$ Using the identity \(\cosh^{2}{t} - 1 = \sinh^{2}{t}\), we can simplify the integral as follows: $$\int \frac{\sinh{t} dt}{\cosh{t} (\sinh^{2}{t})^{3 / 2}} = \int \frac{dt}{\cosh{t} \sinh^2{t}}$$
03

Simplify the integral further and integrate

To simplify the integral further, we use the property \(\sinh^2{t}=\cosh^2{t}-1\). The integral becomes: $$\int \frac{dt}{\cosh{t}\left(\cosh^2{t}-1\right)} = \int \frac{dt}{\cosh^3{t} - \cosh{t}}$$ Now, let \(u=\cosh{t}\), so \(du=\sinh{t} dt\): $$\int \frac{dt}{\cosh^3{t} - \cosh{t}} = \int \frac{du}{u^3 - u}$$ Now, we can factor out the denominator: $$\int \frac{du}{u(u^2 - 1)} = \int \frac{du}{u(u-1)(u+1)}$$
04

Perform partial fraction decomposition and integrate

We will now perform the partial fraction decomposition to get: $$\frac{1}{u(u-1)(u+1)} = \frac{A}{u} + \frac{B}{u-1} + \frac{C}{u+1}$$ Solving for \(A\), \(B\), \(C\), we find that \(A = -\frac{1}{2}\), \(B = \frac{1}{2}\), and \(C = 0\). Then the integral becomes: $$\int\frac{du}{u(u-1)(u+1)} = -\frac{1}{2}\int \frac{du}{u}+\frac{1}{2}\int \frac{du}{u-1}$$ Integrating both terms gives: $$-\frac{1}{2}\ln{|u|} + \frac{1}{2}\ln{|u-1|} + C$$
05

Reverse the substitution

Now, remember that \(u = \cosh{t}\) and \(x = \cosh{t}\). So, the integral becomes: $$-\frac{1}{2}\ln{|\cosh{t}|} + \frac{1}{2}\ln{|\cosh{t}-1|} + C$$ Substituting \(x\) back in, we have: $$-\frac{1}{2}\ln{|x|} + \frac{1}{2}\ln{|x-1|} + C$$ Thus, the value of the integral is: $$\int \frac{dx}{x\left(x^2-1\right)^{3 / 2}} = -\frac{1}{2}\ln{|x|} + \frac{1}{2}\ln{|x-1|} + C$$

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