To solve the integral $\int \sin(x) \log(\tan(x)) \, dx$, we can apply integration by parts.

We will first need to use the formula for integration by parts:

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

Step 1: Choose $u$ and $dv$

We can set:

$u = \log(\tan(x)), \quad dv = \sin(x) \, dx$

Now, differentiate $u$ and integrate $dv$:

$du = \frac{d}{dx} \log(\tan(x)) = \frac{1}{\tan(x)} \cdot \sec^2(x) \, dx = \frac{\sec^2(x)}{\tan(x)} \, dx$ $v = \int \sin(x) \, dx = -\cos(x)$

Step 2: Apply the Integration by Parts Formula

Now, applying the formula:

$\int \sin(x) \log(\tan(x)) \, dx = -\cos(x) \log(\tan(x)) - \int \left(-\cos(x)\right) \cdot \frac{\sec^2(x)}{\tan(x)} \, dx$

This simplifies to:

$\int \sin(x) \log(\tan(x)) \, dx = -\cos(x) \log(\tan(x)) + \int \frac{\cos(x) \sec^2(x)}{\tan(x)} \, dx$

Step 3: Simplify the Remaining Integral

Now, simplify the second integral:

$\frac{\cos(x) \sec^2(x)}{\tan(x)} = \frac{1}{\tan(x)} \cdot \sec(x)$

Thus, the remaining integral becomes:

$\int \frac{1}{\tan(x)} \cdot \sec(x) \, dx$

We can use a substitution method here. Let $u = \tan(x)$, then $du = \sec^2(x) \, dx$. The integral becomes:

$\int \frac{1}{u} \, du = \log|u| = \log|\tan(x)|$

Step 4: Final Answer

Substituting back into our original equation, we get:

$\int \sin(x) \log(\tan(x)) \, dx = -\cos(x) \log(\tan(x)) + \log|\tan(x)| + C$

where $C$ is the constant of integration.

Final Answer:

$\boxed{-\cos(x) \log(\tan(x)) + \log|\tan(x)| + C}$

Would you like a more detailed explanation of any of the steps? Let me know if you need further clarification!

Here are some related questions:

1. How do we handle the integral of trigonometric functions like $\sin(x)$ and $\cos(x)$ in more complex integrals?
2. How can substitution help simplify integrals involving trigonometric functions?
3. What are some common tricks for dealing with logarithmic integrals?
4. Can integration by parts always be applied to integrals with a product of functions?
5. How do we verify if the result of an integral is correct?

Tip: Always check if the function inside the integral is easy to differentiate or integrate before applying integration by parts!