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Chapter 8: Problem 31

Use integration tables to find the integral. $$ \int \frac{1}{x^{2} \sqrt{2+9 x^{2}}} d x $$

Short Answer

Expert verified
The solution to the integral is \(-\sqrt{\frac{3}{2}} \cdot coth^{-1} \left(\frac{3x}{\sqrt{2}}\right) + C\)

Step by step solution

01

Identification

Identify a standard form integral that can be resulted from the given one via algebraic manipulation. Notice that the denominator is in the form of a product between \(x^2\) and \(\sqrt{2+9x^2}\). It would be helpful if we rewrite the integral in the following form: \(\int \frac{1}{x^2} \cdot \frac{1}{\sqrt{9x^2+2}} \, dx\). We can thus identify possible pattern of inverse hyperbolic functions.
02

Substitution

With the form identified, let's proceed with substitution to simplify the integral into the standard form identified: Now, choose the substitution as \(u= 3x\) such that \(du=3dx\), which will give us the integral in this form: \(\frac{1}{3}\int \frac{1}{u^2} \cdot \frac{1}{\sqrt{u^2+\frac{2}{3}}} du\)
03

Integration

Now, this integral is recognisable in the form of \(\int \frac{1}{u^2} \cdot \frac{1}{\sqrt{u^2+a^2}} du\), using the properties of inverse hyperbolic function, it can be solved as \(-\frac{1}{a} \cdot coth^{-1} \left(\frac{u}{a}\right) + C\). Here, \(a = \sqrt{\frac{2}{3}}\). From this, we get the integral as \(-\sqrt{\frac{3}{2}} \cdot coth^{-1} \left(\frac{3u}{\sqrt{2}}\right) + C\).
04

Back substitution

Returning to the original variable, we get the final answer in terms of x: \(-\sqrt{\frac{3}{2}} \cdot coth^{-1} \left(\frac{3x}{\sqrt{2}}\right) + C\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Tables
Integration tables are a handy resource for finding integrals quickly without delving into complex calculations. These tables provide a compiled list of common integral forms and their solutions.

This tool is beneficial:
  • When you recognize the integral matches a form in the table.
  • For complex expressions that standard techniques can't tackle easily.
  • To save time when dealing with standard integral forms.
In this exercise, the integral \(\int \frac{1}{x^{2} \sqrt{2+9 x^{2}}} \, dx\)was initially manipulated and identified to match a standard form, allowing the integral to be solved using inverse hyperbolic functions from the tables.
Inverse Hyperbolic Functions
Inverse hyperbolic functions are analogs of trigonometric functions, useful in solving integrals involving expressions of square roots and squares.

When encountering integrals like this, the inverse hyperbolic expressions become useful:
  • They often simplify complex integrals into solvable forms using known algebraic identities.
  • The integral \(\int \frac{1}{u^2} \cdot \frac{1}{\sqrt{u^2+a^2}} du\) is solved using \(-\frac{1}{a} \cdot \text{coth}^{-1} \left(\frac{u}{a}\right) + C\).
  • In our solution, \(a = \sqrt{\frac{2}{3}}\), which allows this transformation.
These functions are ideal for situations where regular trigonometric identities don鈥檛 apply and provide a pathway to back-substitute into the original variable.
Substitution Method
The substitution method is a technique where you simplify an integral using a change of variables, making it easier to integrate.

This process aids:
  • When an integral seems complex, substitution often re-casts it into an easier form.
  • It involves picking a substitution that matches or resembles a standard integral.
  • For example, setting \(u = 3x\) transformed \(\int \frac{1}{x^{2} \sqrt{2+9 x^{2}}} \, dx\) into a more manageable integral \(\frac{1}{3}\int \frac{1}{u^2} \cdot \frac{1}{\sqrt{u^2+\frac{2}{3}}} du\).
Finally, the back substitution converts our solved integral in terms of \(u\) back to \(x\), yielding the final answer. This technique is crucial in bridging complex integrals to solvable forms.

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Most popular questions from this chapter

Finding a Pattern (a) Find \(\int \cos ^{3} x d x\). (b) Find \(\int \cos ^{5} x d x\). (c) Find \(\int \cos ^{7} x d x\). (d) Explain how to find \(\int \cos ^{15} x d x\) without actually integrating.

Apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that $$\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$ Functions \(\quad\) Interval $$ f(x)=x^{3}, \quad g(x)=x^{2}+1 $$ $$ [0,1] $$

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The region bounded by \((x-2)^{2}+y^{2}=1\) is revolved about the \(y\) -axis to form a torus. Find the surface area of the torus.

For what value of \(c\) does the integral \(\int_{0}^{\infty}\left(\frac{1}{\sqrt{x^{2}+1}}-\frac{c}{x+1}\right) d x\) converge? Evaluate the integral for this value of \(c\).
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