https://file.aiquickdraw.com/mathbot/file/1726183403073e74aq0ld.jpg

The integral you uploaded is:

\[
\int \sqrt{\cos(\theta)} \sin^3(\theta) \, d\theta
\]

### Solution Strategy:
1. **Substitution method**: We can use the trigonometric identity and substitutions to simplify the integral.

Let's start by setting:
\[
u = \cos(\theta)
\]
This gives:
\[
du = -\sin(\theta) d\theta
\]

Also, notice that:
\[
\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - u^2
\]

Now, we can rewrite \(\sin^3(\theta)\) as:
\[
\sin^3(\theta) = \sin(\theta) \cdot \sin^2(\theta) = \sin(\theta) (1 - \cos^2(\theta)) = \sin(\theta) (1 - u^2)
\]

Thus, the integral becomes:
\[
\int \sqrt{u} \cdot \sin(\theta)(1 - u^2) \, d\theta
\]
Now substitute \( du = -\sin(\theta) d\theta \), yielding:
\[
-\int \sqrt{u} (1 - u^2) \, du
\]

Now, distribute and simplify:
\[
-\int \left( u^{1/2} - u^{5/2} \right) \, du
\]

This gives us two simpler integrals to solve:
\[
-\left( \int u^{1/2} \, du - \int u^{5/2} \, du \right)
\]

Now, integrate each term:
\[
-\left( \frac{2}{3} u^{3/2} - \frac{2}{7} u^{7/2} \right) + C
\]

Finally, substitute \( u = \cos(\theta) \) back in:
\[
-\left( \frac{2}{3} \cos^{3/2}(\theta) - \frac{2}{7} \cos^{7/2}(\theta) \right) + C
\]

Thus, the solution is:
\[
-\frac{2}{3} \cos^{3/2}(\theta) + \frac{2}{7} \cos^{7/2}(\theta) + C
\]

Would you like further details on any step, or do you have any questions?

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Here are some related questions:
1. How do substitution methods simplify complex trigonometric integrals?
2. What are other possible methods for solving trigonometric integrals?
3. Could partial fractions be used in integrals like this one?
4. How would this integral change if the powers of sine and cosine were different?
5. How does one check the result of an integral?

**Tip**: When solving trigonometric integrals, always look for identities or substitutions that reduce the powers of sine and cosine.

Integrate \( \sqrt{\cos(\theta)} \sin^3(\theta) \, d\theta \).

Solve the integral of \( \sqrt{\cos(\theta)} \sin^3(\theta) \) step-by-step using substitution. Learn the trigonometric identities and substitution method to simplify and compute complex integrals. Suitable for Grades 11-12.

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