In the following exercises, find the antiderivative using the
indicated substitution.∫ cos3 θdθ; u = sinθ (Hint: cos2 θ = 1 − sin2 θ)

To solve the integral \(\int \cos^3 \theta \, d\theta\) using the substitution \(u = \sin\theta\), we'll proceed step by step.

### Step 1: Express the integral in terms of \(u\)
Given the substitution \(u = \sin\theta\), we can differentiate \(u\) with respect to \(\theta\):
\[
du = \cos\theta \, d\theta
\]

Now, we rewrite the integral:
\[
\int \cos^3 \theta \, d\theta = \int \cos^2 \theta \cdot \cos\theta \, d\theta
\]

Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\), we substitute:
\[
\cos^2 \theta = 1 - u^2
\]

Thus, the integral becomes:
\[
\int (1 - u^2) \cos\theta \, d\theta
\]

Since \(du = \cos\theta \, d\theta\), we substitute:
\[
\int (1 - u^2) \, du
\]

### Step 2: Integrate with respect to \(u\)
Now, integrate each term:
\[
\int (1 - u^2) \, du = \int 1 \, du - \int u^2 \, du
\]

Integrating each part:
\[
\int 1 \, du = u
\]
\[
\int u^2 \, du = \frac{u^3}{3}
\]

So, the integral becomes:
\[
u - \frac{u^3}{3} + C
\]

### Step 3: Substitute \(u\) back in terms of \(\theta\)
Recall that \(u = \sin\theta\). Substituting back, we get:
\[
\sin\theta - \frac{\sin^3\theta}{3} + C
\]

### Final Answer
The antiderivative is:
\[
\int \cos^3 \theta \, d\theta = \sin\theta - \frac{\sin^3\theta}{3} + C
\]

Would you like more details or have any questions? 

Here are 5 related questions you might find useful:
1. How would you find the antiderivative of \(\int \cos^5 \theta \, d\theta\) using substitution?
2. What is the general form for integrating powers of cosine, \(\int \cos^n \theta \, d\theta\)?
3. How can you verify the result by differentiating the antiderivative \(\sin\theta - \frac{\sin^3\theta}{3} + C\)?
4. What would happen if we chose a different substitution, such as \(u = \cos\theta\)?
5. How do trigonometric identities play a role in simplifying integrals?

**Tip:** When dealing with trigonometric integrals, always look for useful identities or substitutions that can simplify the integrand.

Learn how to find the antiderivative of cos^3(theta) using the substitution u = sin(theta). This problem involves integration and trigonometric identities, suitable for advanced high school students and above.

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