integral left parenthesis four times x minus three right parenthesis times sine of three times x d x equals

To find the integral \(\int (4x - 3) \sin(3x) \, dx\), we can use integration by parts. Let's proceed step by step.

### Step 1: Identify \( u \) and \( dv \)
We can choose:
\[
u = 4x - 3 \quad \text{and} \quad dv = \sin(3x) \, dx
\]

### Step 2: Differentiate \( u \) and integrate \( dv \)
Find \( du \) by differentiating \( u \) with respect to \( x \):
\[
du = 4 \, dx
\]
Now, integrate \( dv \) to find \( v \):
\[
v = \int \sin(3x) \, dx = -\frac{1}{3} \cos(3x)
\]

### Step 3: Apply the integration by parts formula
The integration by parts formula is:
\[
\int u \, dv = uv - \int v \, du
\]
Substituting the values we found:
\[
\int (4x - 3) \sin(3x) \, dx = (4x - 3) \left(-\frac{1}{3} \cos(3x)\right) - \int \left(-\frac{1}{3} \cos(3x)\right) \cdot 4 \, dx
\]

### Step 4: Simplify and solve the remaining integral
Simplify the expression:
\[
\int (4x - 3) \sin(3x) \, dx = -\frac{4x - 3}{3} \cos(3x) + \frac{4}{3} \int \cos(3x) \, dx
\]
Now, integrate \( \cos(3x) \):
\[
\int \cos(3x) \, dx = \frac{1}{3} \sin(3x)
\]
Substitute back:
\[
\int (4x - 3) \sin(3x) \, dx = -\frac{4x - 3}{3} \cos(3x) + \frac{4}{9} \sin(3x) + C
\]
where \( C \) is the constant of integration.

### Final Answer
\[
\boxed{\int (4x - 3) \sin(3x) \, dx = -\frac{4x - 3}{3} \cos(3x) + \frac{4}{9} \sin(3x) + C}
\]

Would you like to see any details or have any questions?

Here are 5 related questions to consider:
1. How do you apply integration by parts in different scenarios?
2. What happens if you choose different functions for \( u \) and \( dv \)?
3. How would the integration process change if it were \( (4x^2 - 3x) \sin(3x) \)?
4. Can you perform the integral using a different method, like substitution?
5. How does the choice of \( u \) and \( dv \) affect the complexity of the integral?

**Tip:** When choosing \( u \) and \( dv \) in integration by parts, pick \( u \) to be the function that becomes simpler when differentiated, and \( dv \) to be the part that can be easily integrated.

Learn how to integrate (4x - 3)sin(3x) step-by-step with a detailed solution. This problem involves using integration by parts and trigonometric integrals. Suitable for advanced high school and college students.

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