Need integral help, i think its trig sub., thanks

In summary, the conversation discusses different substitution methods for the integral \int_{0}^{a}\frac{x}{(x^2+z^2)^\frac{3}{2}}dx, with the final agreed upon substitution being t= x^2+ z^2. This results in a simpler integral of \frac{1}{2}\int_{z^2}^{z^2+ a^2} t^{-3/2}dt.
  • #1
klawlor419
117
0
[tex]\int_{0}^{a}\frac{x}{(x^2+z^2)^\frac{3}{2}}dx[/tex]

Hi all, I'm stuck on this one. Sure there's an easier way to do it. Don't have my calc book currently. Thanks!
 
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  • #2
Put x2 = t.
 
  • #3
Got it with the trig sub [tex]x=z \tan{\theta}[/tex]
Thanks!
 
  • #4
klawlor419 said:
Got it with the trig sub [tex]x=z \tan{\theta}[/tex]
Thanks!

No, that substitution isn't convenient,

If you substitute x2 = t, you would get the numerator as dt (since dt= 2xdx)
 
  • #5
Thanks! That is slightly easier!
 
  • #6
I would have used [itex]t= x^2+ z^2[/itex] so that [itex]dt= 2x dx[/itex], [itex](1/2)dt= xdx[/itex].

The integral becomes
[tex]\frac{1}{2}\int_{z^2}^{z^2+ a^2} t^{-3/2}dt[/tex]
 
  • #7
Thanks! Thats slightly easier still!
 

1. What is a trigonometric substitution?

A trigonometric substitution is a method used in calculus to evaluate integrals involving expressions with trigonometric functions. It involves substituting a trigonometric function, such as sine or cosine, for a variable in the integral.

2. How do I know when to use trigonometric substitution?

Trigonometric substitution is typically used when the integrand contains a combination of radicals and trigonometric functions. It is also useful for integrals involving expressions with squares of trigonometric functions.

3. Can I use any trigonometric function for substitution?

Yes, you can use any of the six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) for substitution. The choice of which function to use depends on the form of the integral.

4. How do I perform a trigonometric substitution?

To perform a trigonometric substitution, you need to choose an appropriate trigonometric function to substitute into the integral. Then, use trigonometric identities to rewrite the integral in terms of the chosen function. Finally, solve the resulting integral using integration techniques.

5. What are some common trigonometric identities used in trigonometric substitution?

Some common trigonometric identities used in trigonometric substitution include:
- sin2(x) + cos2(x) = 1
- 1 + tan2(x) = sec2(x)
- 1 + cot2(x) = csc2(x)
- sin(x) = cos(x)tan(x)
- cos(x) = sin(x)/tan(x)
- tan(x) = sin(x)/cos(x)

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