Integral with trig substitution

In summary, the problem is to find the integral of (x^3)/\sqrt{x^2-9} using trig substitution. The solution involves using the substitution x=3*sec\theta and simplifying the integral to get 27\int (1+\tan^2 \theta)\sec^2 \theta d\theta. The final solution is (\frac{1}{3}x^2\,+\,6)\sqrt{x^{2}-9}\,.
  • #1
paralian
14
0
[SOLVED] Integral with trig substitution

Homework Statement



Find [tex]\int(x^3)/\sqrt{x^2-9}[/tex]

Homework Equations



Trig substitution. sin^2 + cos^2 =1, and other things that you can figure out from that.

Half angle formula, cos^2[tex]\theta[/tex]=(1+cos(2[tex]\theta[/tex]) )*.5

The Attempt at a Solution



Let x=3*sec[tex]\theta[/tex]
so dx=3*sec[tex]\theta[/tex]*tan[tex]\theta[/tex] d[tex]\theta[/tex]

When I substitute that in and simplify it, I got:

27*[tex]\int(sec^4(\theta) d\theta)[/tex]

And I don't know how to integrate that. Half angle formulas aren't seeming to work.

Thanks!
 
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  • #2
Or, you can transform some of the secants into tangents:

[tex]27\int \sec^4 \theta d\theta = 27\int (1+\tan^2 \theta)\sec^2 \theta d\theta[/tex]

and you can go from there.
 
  • #3
Tedjn said:
Or, you can transform some of the secants into tangents:

[tex]27\int \sec^4 \theta d\theta = 27\int (1+\tan^2 \theta)\sec^2 \theta d\theta[/tex]

and you can go from there.

[tex]27\int (1+\tan^2 \theta)\sec^2 \theta d\theta[/tex]

[tex]27\int \sec^2 \theta d\theta + 27\int \tan^2 \theta\sec^2 \theta d\theta[/tex]

[tex]27*tan\theta + 9\tan^3 \theta[/tex]

Haha sorry I kind of forgot that [tex]\int \sec^2\theta d\theta[/tex] was tan!
 
  • #4
You are almost done. Because the problem was originally in terms of x, you will need to transform [itex]\theta[/itex] back into x.
 
  • #5
There's no need for trigonometry. Part integration and an immediate substitution will do the trick.

[tex] \int \frac{x^{3}}{\sqrt{x^{2}-9}}dx=\int x^{2}\frac{x}{\sqrt{x^{2}-9}}dx=\allowbreak x^{2}\sqrt{x^{2}-9}-2\int x\sqrt{x^{2}-9}\,dx=\allowbreak x^{2}\sqrt{x^{2}-9}-\frac{2}{3}\left( \sqrt{x^{2}-9}\right) ^{3} [/tex]

up to a constant of integration.
 
  • #6
… just simplify …

Integration by parts not necessary:

[tex] \int \frac{x^{3}}{\sqrt{x^{2}-9}}dx
=\int x\sqrt{x^{2}-9}dx\,+\,\int \frac{9x}{\sqrt{x^{2}-9}}dx
\,=\,\frac{1}{3}(x^{2}-9)^{3/2}\,+\,9\sqrt{x^{2}-9}
\,=\,(\frac{1}{3}x^2\,+\,6)\sqrt{x^{2}-9}\,.[/tex]
 
  • #7
Polynomial division?
 
  • #8
Tedjn said:
You are almost done. Because the problem was originally in terms of x, you will need to transform [itex]\theta[/itex] back into x.

Right...I always forget that!

[tex]x=3*sec \theta \Rightarrow \theta = Sec^-1 (x/3)[/tex]

[tex]\Rightarrow tan\theta = \sqrt{x^2-9}/3[/tex]

[tex]9 \sqrt{x^2-9} + (x^2-9)^\frac{3}{2} /3[/tex]

Shiny! Thanks
 

What is "Integral with trig substitution"?

"Integral with trig substitution" is a technique used in calculus to solve integrals involving trigonometric functions. It involves replacing the variable of integration with a trigonometric function to simplify the integral and make it easier to solve.

Why is trig substitution used in integrals?

Trig substitution is used in integrals because it allows for the simplification of the integral and makes it easier to solve. It also allows for the integration of functions that cannot be solved using other techniques.

How do you perform trig substitution in an integral?

To perform trig substitution in an integral, you first identify the appropriate trigonometric function to substitute for the variable of integration. Then, you use trigonometric identities to rewrite the integral in terms of the substituted variable. Finally, you solve the integral using standard integration techniques.

What are the common trigonometric substitutions used in integrals?

The three most common trigonometric substitutions used in integrals are:

  • Sine substitution: used when the integral contains expressions of the form √(a^2 - x^2)
  • Cosine substitution: used when the integral contains expressions of the form √(a^2 + x^2)
  • Tangent substitution: used when the integral contains expressions of the form a^2 + x^2

What are some tips for using trig substitution effectively?

Some tips for using trig substitution effectively include:

  • Choose the appropriate trigonometric substitution based on the expression in the integral
  • Always check your answer by differentiating it to ensure it is correct
  • Be familiar with trigonometric identities and know when to use them
  • Practice solving different types of integrals using trig substitution to improve your skills

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