What is the integral of sin^3 theta?

So it would be sin theta - (1/3)sin^3 theta + CIn summary, the conversation discusses finding the integral of cos^3 theta and provides a hint on how to approach it by using the substitution cos theta = (1-t^2)/(1+t^2). After some calculations, the correct answer is determined to be sin theta - (1/3)sin^3 theta + C.
  • #1
redshift
53
0
Would very much appreciate a hint on how to start this off.
So far I have cos theta = (1-t^2)/(1+t^2). After squaring this and doing various substitutions, I get (cos theta)^3, which can't be right.
 
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  • #2
I'm not sure what you're asking but if you want the integral indicated in the title of your post this may help:

[tex]\cos^3 \theta = \cos^2 \theta \cos \theta = \left(1-\sin^2 \theta \right) \frac {d \sin \theta}{d\theta}[/tex]
 
  • #3
Many thanks. I'll give it a shot.
 
  • #4
I got cos theta - (1/3)cos^3 theta + C
Does this look right?
 
  • #5
Almost. It should be sin, not cos.
 

What is the integral of cos^3 theta?

The integral of cos^3 theta is equal to (sin theta * cos^2 theta)/3 + (2/3) * integral of cos theta d(theta).

How do you solve the integral of cos^3 theta?

To solve the integral of cos^3 theta, you can use integration by parts or the trigonometric substitution method.

What is the difference between the integral of cos^3 theta and the integral of sin^3 theta?

The main difference between the integral of cos^3 theta and the integral of sin^3 theta is that the former includes an extra cosine term, making the integration more complex.

What is the domain of the integral of cos^3 theta?

The domain of the integral of cos^3 theta is all real numbers.

What applications does the integral of cos^3 theta have in real life?

The integral of cos^3 theta is commonly used in physics and engineering to calculate the work done by a variable force acting on a particle along a curved path.

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