To evaluate the integral $\int x \sin\left(\frac{1}{7}x\right) \, dx$ using integration by parts, we start by applying the formula for integration by parts:

$\int u \, dv = u v - \int v \, du$

Step 1: Assign $u$ and $dv$

We choose:

• $u = x$, so $du = dx$
• $dv = \sin\left(\frac{1}{7}x\right) \, dx$, so we need to find $v$.

Step 2: Compute $v$

We integrate $dv = \sin\left(\frac{1}{7}x\right) \, dx$:

$v = \int \sin\left(\frac{1}{7}x\right) \, dx = -7 \cos\left(\frac{1}{7}x\right)$ (The factor of 7 comes from the chain rule when integrating.)

Step 3: Apply the integration by parts formula

Now, we apply the formula:

$\int x \sin\left(\frac{1}{7}x\right) \, dx = u v - \int v \, du$

Substituting $u = x$, $du = dx$, $v = -7 \cos\left(\frac{1}{7}x\right)$, we get:

$\int x \sin\left(\frac{1}{7}x\right) \, dx = -7x \cos\left(\frac{1}{7}x\right) - \int -7 \cos\left(\frac{1}{7}x\right) \, dx$

This simplifies to:

$\int x \sin\left(\frac{1}{7}x\right) \, dx = -7x \cos\left(\frac{1}{7}x\right) + 7 \int \cos\left(\frac{1}{7}x\right) \, dx$

Step 4: Evaluate the remaining integral

We now need to compute $\int \cos\left(\frac{1}{7}x\right) \, dx$:

$\int \cos\left(\frac{1}{7}x\right) \, dx = 7 \sin\left(\frac{1}{7}x\right)$

Step 5: Final answer

Substituting this back into our equation:

$\int x \sin\left(\frac{1}{7}x\right) \, dx = -7x \cos\left(\frac{1}{7}x\right) + 49 \sin\left(\frac{1}{7}x\right) + C$

where $C$ is the constant of integration.

Part 1: Choosing the correct answer

Among the choices given:

• The correct answer corresponds to Option D, because it matches the structure of the integration by parts we performed.

Part 2: The final evaluated integral

$\boxed{-7x \cos\left(\frac{1}{7}x\right) + 49 \sin\left(\frac{1}{7}x\right) + C}$

Let me know if you need any further details or explanations!

5 related questions:

1. How do you choose $u$ and $dv$ in integration by parts?
2. What is the chain rule in integration, and how does it apply here?
3. Can integration by parts be applied to trigonometric functions other than sine and cosine?
4. What happens when you apply integration by parts repeatedly?
5. How would this integral change if the limits of integration were specified?

Tip: When using integration by parts, always choose $u$ as the part of the integrand that simplifies upon differentiation, and $dv$ as the part that can be easily integrated.