Integrate sinx(cosx)^2dx using the substitution u=sinx. I know how to do this using u =cosx, but not sinx. The next problem on the homework was the same question except it asked to use u=cosx, so there couldn't have been a mistake.

u = sinx

du = cosx dx
so, dx = du/cosx = du/√(1-u^2)

sinx(cosx)^2 dx = u(1-u^2)/√(1-u^2) du
= u/√(1-u^2) du

Now, let v = √(1-u^2)
dv = -u/√(1-u^2) du

and you have

-v dv

integrate that to get

-1/2 v^2 = -1/2(1-u^2) = -1/2 (1-sin^2(x)) = -1/2 cos^2(x) + C

Now, that probably looks different from what you got letting u=cosx, but I think if you manipulate things, you'll find that the difference is caused by having a different +C at the end.

Or, maybe I made a mistake above...

ok, I got it, it worked out for me to be the same thing as when I used u=cosx

To integrate the function sin(x) * (cos(x))^2, you can indeed use the trigonometric substitution u = cos(x). However, if your homework specifically asks you to use the substitution u = sin(x), it means that there is a different approach to solving the problem.

Let's go through the steps of finding the integral using the substitution u = sin(x):

1. Start by differentiating the substitution u = sin(x) with respect to x: du/dx = cos(x).

2. Rearrange the equation to solve for dx: dx = du / cos(x).

3. Substitute sin(x) with u in the original integral:

∫ sin(x) * (cos(x))^2 dx = ∫ (u) * (cos(x))^2 (du / cos(x)).

4. Simplify the integral:

∫ u * (cos(x))^2 (du / cos(x))

Since cos(x) / cos(x) cancels out, this simplifies to:

∫ u * (cos(x))^2 du

5. Focus on the remaining expression: ∫ u * (cos(x))^2 du. Notice that (cos(x))^2 is a constant with respect to u, so it can be moved outside the integral:

∫ (cos(x))^2 * u du

6. Now, integrate the expression as a simple polynomial:

∫ (cos(x))^2 * u du = (cos(x))^2 * (u^2 / 2) + C,

where C is the constant of integration.

7. Replace u with sin(x):

(cos(x))^2 * (u^2 / 2) + C = (cos(x))^2 * (sin(x))^2 / 2 + C.

Therefore, the integral of sin(x) * (cos(x))^2 using the substitution u = sin(x) is given by (cos(x))^2 * (sin(x))^2 / 2 + C, where C is the constant of integration.

However, if your homework specifically states to use u = cos(x) instead, you can follow the same steps using that substitution to find the integral.