https://file.aiquickdraw.com/mathbot/file/1727657924047yalj3bvp.jpg

The integral provided is:

\[
\int 2 \sin^4(x) \cos(x) \, dx
\]

To solve this, we can use substitution and trigonometric identities.

### Step 1: Substitution
Let:
\[
u = \sin(x)
\]
Then:
\[
\frac{du}{dx} = \cos(x) \quad \Rightarrow \quad du = \cos(x) \, dx
\]

### Step 2: Rewrite the integral
Using the substitution \( u = \sin(x) \), the integral becomes:
\[
\int 2 u^4 \, du
\]

### Step 3: Integrate
Now integrate the expression:
\[
\int 2 u^4 \, du = 2 \cdot \frac{u^5}{5} = \frac{2}{5} u^5
\]

### Step 4: Substitute back \( u = \sin(x) \)
Now, replace \( u \) with \( \sin(x) \):
\[
\frac{2}{5} \sin^5(x)
\]

Thus, the solution to the integral is:

\[
\frac{2}{5} \sin^5(x) + C
\]

Let me know if you'd like further details or explanations!

Here are some related questions:
1. How does substitution help in evaluating integrals?
2. Can you provide an example where the substitution method does not work well?
3. How would you approach an integral with higher powers of sine and cosine?
4. How does integration differ from differentiation in terms of complexity?
5. How can you verify the result of an indefinite integral?

**Tip**: Always double-check your substitution and integration limits (for definite integrals) to avoid errors when reversing substitutions!

Evaluate the indefinite integral: ∫ 2sin⁴(x)cos(x)dx

Learn how to solve the indefinite integral ∫ 2sin⁴(x)cos(x)dx using substitution. This detailed step-by-step guide explains the substitution method, key trigonometric identities, and the power rule for integration. Suitable for students in Grades 11-12 or early college-level calculus.

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