Question

With clean steps and answers, evaluate the indefinite integral: ∫sqrt(cos(x)) sin^3(x) dx.

Answer 1

The indefinite integral ∫ sqrt(cos(x)) sin^3(x) dx can be solved using substitution. Let u = cos(x), then integrate the transformed expression ∫ -u^(1/2) (1 - u^2) du, and substitute back to obtain the final answer in terms of x.

Step-by-step explanation:

To evaluate the indefinite integral ∫ sqrt(cos(x)) sin^3(x) dx, we can use a substitution method that takes advantage of the relationship between sine and cosine functions. Recall that sin^2(x) + cos^2(x) = 1 and that the derivatives of sine and cosine are related. We can let u = cos(x), which means that du = -sin(x) dx. This substitution simplifies the integral as follows:

1. Let u = cos(x). Then du = -sin(x) dx.
2. The integral becomes ∫ sqrt(u) (-sin^2(x)) du.
3. Replace sin^2(x) with 1 - u^2, since sin^2(x) + cos^2(x) = 1.
4. The integral now is ∫ -u^(1/2) (1 - u^2) du.
5. Expand and integrate term by term.
6. Finally, substitute back u = cos(x) to obtain the answer in terms of x.

This method allows us to transform the integral into a more manageable form that can be integrated using basic integration techniques.