Question

Differentiate (sin^3)x and use this result to evaluate the integral of sin^2 x cos x dx between pi/2 and 0.

Thanks!

Answer 1

Well, the derivative is 3 sin^2x cosx. You get that using the "chain rule"

That means the derivative of (sin^3x)/3 is sin^2x cosx

So the indefinite integral of sin^2x cosx is sin^3x/3

For the definite integral, subtract sin^3(pi/2)/3 from sin^3(0)/3
The answer is -1/3, assuming the integral goes backwards from pi/2 to zero, as you seem to imply.
If the integration is in the forward direction, the answer is 1/3.