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Chapter 4: Problem 10

Find the integral. $$ \int \frac{1}{x \sqrt{x^{4}-4}} d x $$

Short Answer

Expert verified
The evaluated integral is \(\frac{1}{2}\ln|x^{2}/\sqrt{x^{4}-4}+ \sqrt{x^{4}/4-1}|\) + C.

Step by step solution

01

Perform a substitution

Let's perform a substitution to simplify this integral. Set \(u = x^2\). This means \(du = 2x \, dx\), and in terms of \(x \, dx\) is \(du/2\). Thus, the original integral now becomes \(\int \frac{du}{2u \sqrt{u^{2}-4}}\).
02

Further simplify the integral

We can simplify the integral by taking out the constant term from the integral and applying the rule \( \int f(x) \, dx = F(x) + C \), which states that the integral of a function is its antiderivative plus a constant. This gives us: \(\frac{1}{2}\int \frac{du}{u \sqrt{u^{2}-4}}\).
03

Use a standard integral formula

This integral is now in the form of a standard integral: \(\int \frac{du}{u \sqrt{u^{2}-a^{2}}}\) where \(a = 2\). The standard integral has the result: \(\frac{1}{a}\ln|u/\sqrt{u^{2}-a^{2}}+ \sqrt{u^{2}/a^{2}-1}|+C\). Therefore, substituting \(a = 2\), the result is \(\frac{1}{2}\ln|u/\sqrt{u^{2}-4}+ \sqrt{u^{2}/4-1}|\).
04

Substitute back

Substitute \(u = x^2\) back, the result is then \(\frac{1}{2}\ln|x^{2}/\sqrt{x^{4}-4}+ \sqrt{x^{4}/4-1}|\).

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Most popular questions from this chapter

From the vertex \((0, c)\) of the catenary \(y=c \cosh (x / c)\) a line \(L\) is drawn perpendicular to the tangent to the catenary at a point \(P\). Prove that the length of \(L\) intercepted by the axes is equal to the ordinate \(y\) of the point \(P\).

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x} t \cos t d t $$

(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{4}^{x} \sqrt{t} d t $$

Linear and Quadratic Approximations In Exercises 33 and 34 use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\). Use a graphing utility to graph the function and its linear and quadratic approximations. \(f(x)=\tanh x, \quad a=0\)

Verify each rule by differentiating. Let \(a>0\). $$ \int \frac{d u}{u \sqrt{u^{2}-a^{2}}}=\frac{1}{a} \operatorname{arcsec} \frac{|u|}{a}+C $$
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