$$ \int\cos\left(3x+2\right)\sin^4\left(3x+2\right)\text{ dx} $$

We are tasked with evaluating the integral:

\[
\int \cos(3x + 2) \sin^4(3x + 2) \, dx
\]

### Step 1: Substitution
Let us start by making the substitution:
\[
u = 3x + 2
\]
This gives:
\[
du = 3 \, dx \quad \Rightarrow \quad dx = \frac{du}{3}
\]

Substituting into the integral:
\[
\int \cos(u) \sin^4(u) \frac{du}{3}
\]

Factoring out the constant:
\[
\frac{1}{3} \int \cos(u) \sin^4(u) \, du
\]

### Step 2: Simplifying with Power Reduction
We can express $\sin^4(u)$ using the identity $\sin^2(u) = 1 - \cos^2(u)$:
\[
\sin^4(u) = (\sin^2(u))^2 = (1 - \cos^2(u))^2
\]

Expanding $(1 - \cos^2(u))^2$:
\[
\sin^4(u) = 1 - 2\cos^2(u) + \cos^4(u)
\]

Thus, the integral becomes:
\[
\frac{1}{3} \int \cos(u) \left(1 - 2\cos^2(u) + \cos^4(u)\right) \, du
\]

### Step 3: Breaking the Integral into Three Terms
We can now split the integral into three simpler integrals:
\[
\frac{1}{3} \left( \int \cos(u) \, du - 2 \int \cos^3(u) \, du + \int \cos^5(u) \, du \right)
\]

#### Integral 1: $\int \cos(u) \, du$
This is straightforward:
\[
\int \cos(u) \, du = \sin(u)
\]

#### Integral 2: $\int \cos^3(u) \, du$
Using the identity $\cos^3(u) = \cos(u)(1 - \sin^2(u))$, we get:
\[
\int \cos^3(u) \, du = \int \cos(u) \, du - \int \cos(u) \sin^2(u) \, du
\]
The first part is $\sin(u)$, while the second part requires a substitution. However, this integral follows a known result:
\[
\int \cos^3(u) \, du = \frac{\sin(u)}{3} + \frac{\sin^3(u)}{3}
\]

#### Integral 3: $\int \cos^5(u) \, du$
Similarly, we use the identity $\cos^5(u) = \cos(u)(1 - \sin^2(u))^2$, and the result for this is:
\[
\int \cos^5(u) \, du = \frac{\sin(u)}{5} + \frac{\sin^3(u)}{5} + \frac{\sin^5(u)}{5}
\]

### Step 4: Combine Results
Now, combining all the terms, we have:

\[
\frac{1}{3} \left( \sin(u) - 2 \left( \frac{\sin(u)}{3} + \frac{\sin^3(u)}{3} \right) + \left( \frac{\sin(u)}{5} + \frac{\sin^3(u)}{5} + \frac{\sin^5(u)}{5} \right) \right)
\]

Finally, substitute $u = 3x + 2$ back into the result.

Would you like me to simplify further or provide details on any of these steps?

---

**Related questions:**
1. How do you solve integrals involving trigonometric identities?
2. What are common techniques for solving trigonometric integrals?
3. Can substitution always be applied for integrals like this?
4. How is the power reduction formula used in integration?
5. What are the differences between definite and indefinite integrals involving trigonometric functions?

**Tip:** Use trigonometric identities to simplify complex integrals involving powers of sine and cosine.

$$ \int \cos\left(3x+2\right)\sin^4\left(3x+2\right)\text{ dx} $$

Learn how to evaluate the integral $$ \int \cos(3x + 2)\sin^4(3x + 2) dx $$ with step-by-step guidance using substitution and power reduction identities. This solution explains core concepts in trigonometric integration, making it suitable for students at the high school calculus or early college level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation