Use trigonometric substitution to verify

In summary, using trigonometric substitution, we can verify that the integral from 0 to pi of the square root of a^2 - t^2 is equal to (1/2)a^2 sin^-1(x/a) + (1/2)x square root of a^2 - x^2. To solve this, we replaced x with a sin(theta) and used a double angle formula to simplify the integral.
  • #1
celeramo
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0

Homework Statement


Using trigonometric substitution, verify that
[tex]\int[/tex] [tex]\sqrt{a^2[-t^2}[/tex]dt (INTEGRAL FROM 0 TO [tex]\pi[/tex])=(1/2)a^2sin^{-1}(x/a)+(1/2)x[tex]\sqrt{a^2-x^2[/tex]

Sorry it doesn't seem to want to allow me to place superscript inside a square root. but inside the first sq. root is a2-t2 and the second sq. root contains a2-x2



Homework Equations



[tex]\sqrt{a^2-x^2}[/tex]

x=asin([tex]\theta[/tex])

-([tex]\pi[/tex]/2) [tex]\leq[/tex] [tex]\theta[/tex] [tex]\leq[/tex] ([tex]\pi[/tex]/2)

The Attempt at a Solution



replacing x with asin[tex]\theta[/tex] in the original integral
[tex]\int[/tex][tex]\sqrt{a^2-(asin^2\theta}[/tex]dt
multiply and factor out. Change 1-sin^2[tex]\theta[/tex] to cos^2[tex]\theta[/tex]
replace dt with dt=(-a)cos[tex]\theta[/tex] d[tex]\theta[/tex] because t=asin[tex]\theta[/tex]
End up with
[tex]\int[/tex] (a)(cos[tex]\theta[/tex])(-a)(cos[tex]\theta[/tex])d[tex]\theta[/tex]
Combine and get
-[tex]\int[/tex] a2cos2[tex]\theta[/tex] d[tex]\theta[/tex]

At this point I considered pulling the a2 out in front since, unless I'm confused, it represents a constant and changing cos2[tex]\theta[/tex] to 1-sin2[tex]\theta[/tex] but that hasn't seemed to get me any close. Any help would be much appreciated or if I've made a mistake already identifying that for me would be excellent. Please and thank you all.
:)
 
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  • #2
No, changing cos^2(theta) to sin^2(theta) doesn't help much. But using a double angle formula does. Change it to (1+cos(2*theta))/2
 
  • #3
Thanks a lot.
 

1. What is trigonometric substitution?

Trigonometric substitution is a technique used in calculus to solve integrals involving expressions with radicals, such as √(a^2 - x^2) or √(x^2 + a^2). It involves substituting the variable x with a trigonometric function, such as sine, cosine, or tangent, to simplify the expression and make it easier to integrate.

2. How do you determine which trigonometric substitution to use?

The substitution you use depends on the expression inside the square root. If the expression is of the form a^2 - x^2, then use x = a sinθ. If the expression is of the form x^2 + a^2, then use x = a tanθ. If the expression is of the form x^2 - a^2, then use x = a secθ. You may also need to use trigonometric identities to simplify the expression further.

3. Can you give an example of using trigonometric substitution to verify an integral?

Sure, let's say we want to verify the integral ∫√(1 - x^2) dx. We can use the substitution x = sinθ. This transforms the integral into ∫√(1 - sin^2θ) cosθ dθ. Using the Pythagorean identity sin^2θ + cos^2θ = 1, we can simplify this to ∫cos^2θ dθ. Using the double angle formula cos^2θ = (1 + cos2θ)/2, we get ∫(1 + cos2θ)/2 dθ. This can be easily integrated to get the answer 1/2 (θ + sinθ cosθ) + C. Substituting back for θ = arcsin(x), we get the final answer 1/2 (arcsin(x) + x√(1 - x^2)) + C.

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

One common mistake is forgetting to substitute back for the original variable after integrating. Another mistake is using the wrong trigonometric substitution, which can lead to incorrect answers. It's also important to be careful with the limits of integration and make sure they are in terms of the new variable, not the original variable.

5. Are there any tips for using trigonometric substitution effectively?

Yes, it's always helpful to have a good understanding of basic trigonometric identities and their derivatives. This will make it easier to simplify the expressions and integrate them. It's also important to practice and familiarize yourself with the different types of trigonometric substitutions, as well as when to use them. Lastly, always be careful and double-check your work to avoid errors.

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