https://file.aiquickdraw.com/mathbot/file/1728324042435j25rmatt.jpg

The integral given in the image is:

\[
\int \sin(\theta) \cos^3(\theta) \, d\theta
\]
with the substitution \( u = \cos(\theta) \).

Let's solve this step by step.

### Step 1: Substitution
We are given \( u = \cos(\theta) \), which implies that \( du = -\sin(\theta) \, d\theta \).

Now, rewrite the integral using the substitution:
\[
\int \sin(\theta) \cos^3(\theta) \, d\theta = - \int u^3 \, du
\]
(Note: The negative sign comes from \( du = -\sin(\theta) \, d\theta \)).

### Step 2: Integration
Now, integrate \( u^3 \):
\[
-\int u^3 \, du = -\frac{u^4}{4} + C
\]
where \( C \) is the constant of integration.

### Step 3: Back-substitute \( u = \cos(\theta) \)
Replace \( u \) with \( \cos(\theta) \) to get the final answer:
\[
-\frac{\cos^4(\theta)}{4} + C
\]

Thus, the solution to the integral is:
\[
\boxed{-\frac{\cos^4(\theta)}{4} + C}
\]

Would you like a further breakdown of any of the steps or have any other questions?

Here are 5 related questions you might want to explore:
1. How can we handle integrals that involve higher powers of trigonometric functions?
2. What other substitutions could be useful for integrals involving trigonometric functions?
3. How does integration by parts compare with substitution for solving trigonometric integrals?
4. Can this method be applied to more complex trigonometric integrals like \( \sin^n(\theta) \cos^m(\theta) \)?
5. What are some common techniques for solving integrals involving products of sine and cosine?

**Tip:** When choosing a substitution, try to simplify the integrand to a form that is easy to integrate (e.g., powers of variables or simple trigonometric functions).

This detailed solution shows how to solve the trigonometric integral ∫ sin(θ) cos^3(θ) dθ using the substitution u = cos(θ). Learn how to apply substitution methods to simplify and integrate trigonometric functions, perfect for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation