Integration via Trigonometric Substitution

In summary, the problem can be solved by using trigonometric substitution and the chain rule to change the integration variable and simplify the integral.
  • #1
Cpt Qwark
45
1

Homework Statement


Evaluate [tex]\int{\frac{x^2}{(1-x^2)^\frac{5}{2}}}dx[/tex] via trigonometric substitution.
You can do this via normal u-substitution but I'm unsure of how to evaluate via trigonometric substitution.

Homework Equations

The Attempt at a Solution


Letting [tex]x=sinθ[/tex],
[tex]\int{\frac{sin^{2}θ}{(1-sin^{2}θ)^\frac{5}{2}}}dθ=\int{\frac{sin^{2}θ}{(cos^{2}θ)^\frac{5}{2}}}dθ[/tex]
but I'm not sure how the working in the answers gets up to [tex]\int{\frac{x^2}{(1-x^2)^\frac{5}{2}}}dx=\int{\frac{sin^{2}θ}{cos^{4}θ}}dθ[/tex].
 
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  • #2
Try to change the last integral by using other trigonometric functions, like ## \tan\theta ## and ## \sec\theta.##
 
  • #3
Oh, I misunderstand what you want to know. I thought you want to calculate the last thing.

When you starts from the beginning, there should be ## dx##, the integral variable. Then, I think you may find what you missed during your calculation.
 
  • #4
So what is wrong with this:
[tex]\int{\frac{sin^{2}θ}{(cos^{2}θ)^\frac{5}{2}}}dθ=
Daeho Ro said:
Oh, I misunderstand what you want to know. I thought you want to calculate the last thing.

When you starts from the beginning, there should be ## dx##, the integral variable. Then, I think you may find what you missed during your calculation.

Yeah I forgot to type that in, anyway it's trig identity I'm kinda having trouble with atm.
 
  • #5
## dx ## cannot change directly ## d\theta ##. They have to connected by some function.
 
  • #6
Daeho Ro said:
## dx ## cannot change directly ## d\theta ##. They have to connected by some function.

Not too sure what you mean by that.
For functions with the form [tex]\sqrt{a^2-x^2}[/tex] you can express them as [tex]x=asinθ[/tex]
 
  • #7
See,

[tex] \int \dfrac{x^2}{(1-x^2)^{5/2}} dx = \int \dfrac{ \sin^2\theta}{(1-\sin^2\theta)^{5/2}} dx = \int \dfrac{\sin^2 \theta}{\cos^5\theta} dx \neq \int \dfrac{\sin^2 \theta}{\cos^5\theta} d\theta.[/tex]
You missed something in the lase step.
 
  • #8
Yeah it was a typo...
 
  • #9
Then, what is ## dx ## as a function of ## \theta ##?
 
  • #10
The denominator is somehow supposed to be [tex]cos^{4}θ[/tex], not [tex]cos^{5}θ[/tex]. That's all I need help with, nothing else.
 
  • #11
Cpt Qwark said:
The denominator is somehow supposed to be [tex]cos^{4}θ[/tex], not [tex]cos^{5}θ[/tex]. That's all I need help with, nothing else.
Yes, I know and you almost reach the final goal.

##dx## have to change as ## d\theta ## because the last integration is in a form ## \int (\cdots) d\theta ##. But as you know, ## dx \neq d\theta ##.

What is ## dx / d\theta ##? It's really strong hint about this problem.
 
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Likes Cpt Qwark
  • #12
I hope you already got the answer for the problem.

The key idea is chain rule. The differentiation of ## x## with respect to ## \theta ## is ## dx/d\theta = \cos\theta ##. Then, the integration will change as
$$ \int \dfrac{\sin^2\theta}{\cos^5\theta} dx = \int \dfrac{\sin^2\theta}{\cos^5\theta} \dfrac{dx}{d\theta} d\theta = \int \dfrac{\sin^2\theta}{\cos^5\theta} \cos \theta d\theta = \int \dfrac{\sin^2\theta}{\cos^4\theta} d\theta.$$
 

FAQ: Integration via Trigonometric Substitution

1. What is trigonometric substitution in integration?

Trigonometric substitution is a technique used in integration where a trigonometric function is substituted for a variable in the integrand. This is often used when the integrand contains an expression that can be simplified using trigonometric identities.

2. When should I use trigonometric substitution?

Trigonometric substitution is typically used when the integrand contains a radical expression, such as √(a²-x²) or √(a²+x²). It can also be used when the integrand contains a quadratic expression, such as a²-x² or a²+x², where a is a constant.

3. How do I choose which trigonometric substitution to use?

The choice of trigonometric substitution depends on the form of the integrand. For expressions of the form √(a²-x²), we use x = a sinθ, while for expressions of the form √(a²+x²), we use x = a tanθ. If the integrand contains a√(x²-a²), we use x = a secθ, and if it contains a√(x²+a²), we use x = a cotθ.

4. What are the most common mistakes when using trigonometric substitution?

One of the most common mistakes when using trigonometric substitution is forgetting to substitute back for the original variable after integrating. Another mistake is using the wrong substitution, which can lead to incorrect results. It is important to carefully identify the form of the integrand before choosing the appropriate substitution.

5. Can I use trigonometric substitution for all integrals?

No, trigonometric substitution is only applicable to certain types of integrals, specifically ones that involve radical or quadratic expressions. It is not suitable for all integrals, and other techniques, such as u-substitution or integration by parts, may need to be used to solve the integral.

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