Integrate: (sin 2x)^3 dx
I can see the answer, but how do I do this?
(sin 2x)^3dx =
(sin 2x)^2 (sin 2x)dx =
-1/2(1-(cos 2x)^2)dcos(2x)
Thanks!
To integrate the function (sin 2x)^3 dx, you can use a technique called substitution. Here's the step-by-step process:
Step 1: Let's start by substituting u = sin 2x.
Taking the derivative of both sides, we have du/dx = 2 cos 2x.
Rearranging for dx, we get dx = du / (2 cos 2x).
Step 2: Now, substitute the expression for dx in terms of du and cos 2x into the integral:
∫ (sin2x)^3 dx = ∫ (sin 2x)^3 (du / 2 cos 2x).
Step 3: Simplify the expression by canceling out common terms:
∫ (1/2) sin^3 2x du.
Step 4: Lastly, we integrate with respect to u:
(1/2) ∫ sin^3 2x du.
Note: The integral of sin^3 2x can be computed using a technique called reduction formula, but it involves multiple steps beyond a simple substitution. However, if you have the answer to the integral, you can proceed to find the result.
If you provide the answer, I can help you verify it and explain the next steps.