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!

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.

To differentiate (sin^3)x, we can use the chain rule.

First, let's rewrite sin^3(x) as (sin(x))^3.

To find the derivative, we multiply the function by the derivative of the inside function, which is sin(x).

So, the derivative of (sin(x))^3 will be 3(sin(x))^2 * cos(x).

Now, let's move on to evaluating the integral of sin^2(x) * cos(x) dx between pi/2 and 0 using the result we obtained.

The integral of sin^2(x) * cos(x) dx can be rewritten as the integral of sin^2(x) d(sin(x)) since sin(x) is the derivative of cos(x).

Using the chain rule, we can integrate sin^2(x) d(sin(x)) as (1/3)(sin^3(x)) + C, where C is the constant of integration.

Now, we can substitute the limits of the integral and evaluate.

At the upper limit, sin(pi/2) is equal to 1, and at the lower limit, sin(0) is equal to 0.

Therefore, the definite integral of sin^2(x) * cos(x) dx between pi/2 and 0 is [(1/3)(sin^3(pi/2))] - [(1/3)(sin^3(0))] = (1/3)(1^3) - (1/3)(0^3) = 1/3.

So, the value of the integral is 1/3.