integrate cos^3x dx

tnk U

http://answers.yahoo.com/question/index?qid=20070907100407AAo2vgu

wow thank u....

what about
integrate
-sinx*cos^(2)x dx

is that
1/3cos^3x?? rigth?

Yes. I answered that in one of your separate posts

To integrate the expression cos^3(x) dx, we can use the technique of trigonometric substitution.

Step 1: Let's start by expressing cos^3(x) in terms of another trigonometric function. We can use the identity cos^2(x) = 1 - sin^2(x) to rewrite it.

cos^3(x) = cos^2(x) * cos(x) = (1 - sin^2(x)) * cos(x)

Step 2: Now, we can make a substitution. Let's set sin(x) = u, so that cos(x) dx = du.

Differentiate both sides of sin(x) = u with respect to x to obtain cos(x) dx = du.

Step 3: Substitute the expression for cos(x) dx and cos^3(x) in terms of u into the integral:

∫ cos^3(x) dx = ∫ (1 - sin^2(x)) * cos(x) dx
= ∫ (1 - u^2) du

Step 4: The integrand is now a basic polynomial that can be easily integrated:

∫ (1 - u^2) du = u - (1/3)u^3 + C

Step 5: Substitute back u = sin(x) into the result:

u - (1/3)u^3 + C = sin(x) - (1/3)sin^3(x) + C

Thus, the final result is ∫ cos^3(x) dx = sin(x) - (1/3)sin^3(x) + C.