Solving Calculus Integral with x=asin(theta)

  • Thread starter iggybaseball
  • Start date
  • Tags
    Calculus
In summary, to solve the given integral, one can substitute z = 2sin(theta) or z = asin(theta) and use the appropriate trigonometric identities to simplify the expression. The resulting integral can then be solved using standard techniques.
  • #1
iggybaseball
57
0
I am having trouble solving this integral:

[tex]\frac{\1}{(4-z^2)^(3/2)}[/tex]

I know that x = asin(theta)
theta = arcsin (x/a)
d(theta) = 1 / sqrt(4-z^2) dz

but then I get stuck. Could someone give me a hand?

ps there should be a number 1 on top of the fraction and the integral has dz after it respectfully. I couldn't get these two in ( It is the first time I used latex) Thanks
 
Physics news on Phys.org
  • #2
Choose z = asin(theta). With regards to your problem, what should you choose a to be? Differentiate z = asin(theta) to find dz = ?

Regards,
George
 
  • #3
It's the same thing but simpler: let z= 2sin(θ) then dz= 2cos(θ).
[tex]\frac{1}{(4-z^)^{\frac{3}{2}}}= \frac{1}{8(1- sin^2(\theta))^{\frac{3}{2}}}[/tex][tex]= \frac{1}{8 cos^3(\theta)}[/tex]
The integral becomes [tex]\frac{1}{8}\integral{\frac{d\theta}{cos^2(\theta)}[/tex].

The way you were doing it works too, of course.
Since [tex]d\theta= \frac{dz}{\sqrt{((4-z^2)}}[/tex] and [tex]\frac{1}{(4-z^2)^{\frac{3}{2}}}= \frac{1}{4-z^2}\frac{1}{\sqrt{4-z^2}}[/tex], that gives the same thing.
 
Last edited by a moderator:

What is the basic concept of solving calculus integral with x=asin(theta)?

The basic concept of solving calculus integral with x=asin(theta) is to use the substitution rule to replace x with asin(theta) and then use trigonometric identities to simplify the integral and solve for the value of theta.

What are the common trigonometric identities used in solving calculus integral with x=asin(theta)?

The common trigonometric identities used in solving calculus integral with x=asin(theta) are sin^2(theta) + cos^2(theta) = 1, 1 + tan^2(theta) = sec^2(theta), and 1 + cot^2(theta) = csc^2(theta).

How do you apply the substitution rule in solving calculus integral with x=asin(theta)?

To apply the substitution rule, you replace the variable x with asin(theta) and also replace dx with acos(theta)d(theta). This allows you to solve the integral in terms of theta instead of x.

What are the steps to solving calculus integral with x=asin(theta)?

The steps to solving calculus integral with x=asin(theta) are:
1. Substitute x with asin(theta) and dx with acos(theta)d(theta)
2. Use trigonometric identities to simplify the integral
3. Integrate the simplified expression
4. Substitute back in for theta and simplify the final answer.

What are some tips for solving calculus integral with x=asin(theta)?

Some tips for solving calculus integral with x=asin(theta) are to carefully apply the substitution rule, use the common trigonometric identities, and check your answer by differentiating it to make sure it matches the original function. It is also helpful to practice and familiarize yourself with different types of trigonometric integrals.

Similar threads

  • Introductory Physics Homework Help
Replies
6
Views
1K
  • Introductory Physics Homework Help
Replies
21
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
527
Replies
7
Views
216
  • Introductory Physics Homework Help
Replies
1
Views
116
  • Introductory Physics Homework Help
Replies
16
Views
986
  • Introductory Physics Homework Help
Replies
6
Views
1K
  • Introductory Physics Homework Help
Replies
9
Views
829
  • Introductory Physics Homework Help
Replies
14
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
925
Back
Top