how can i integrate (x)log(sinx) dx from 0 to pi

please give full stepwise solution

To integrate the function \( x \log(\sin x) \) from \( 0 \) to \( \pi \), we will use integration by parts, a technique that allows us to transform the integral of a product of functions into a simpler form. Let's break down the process step by step.

Step 1: Set Up Integration by Parts

Integration by parts is based on the formula:

∫ u dv = uv - ∫ v du

For our integral, we can choose:

• u = x (which makes it easier to differentiate)
• dv = \log(\sin x) \, dx (which we will need to integrate)

Step 2: Differentiate and Integrate

Now, we need to find \( du \) and \( v \):

• du = dx
• To find \( v \), we need to integrate \( dv \):
• v = ∫ \log(\sin x) \, dx

Finding \( v \) directly can be quite complex, but we will address that later. For now, let’s proceed with the integration by parts formula.

Step 3: Apply the Integration by Parts Formula

Using the integration by parts formula, we have:

∫_{0}^{\pi} x \log(\sin x) \, dx = \left[ x \cdot v \right]_{0}^{\pi} - ∫_{0}^{\pi} v \, dx

Next, we need to evaluate \( \left[ x \cdot v \right]_{0}^{\pi} \). However, note that \( \log(\sin x) \) approaches \(-\infty\) as \( x \) approaches \( 0 \) or \( \pi \). Thus, we must be cautious with this boundary evaluation.

Step 4: Evaluate the Boundary Terms

At \( x = 0 \):

As \( x \to 0 \), \( \sin x \to 0 \), hence \( \log(\sin x) \to -\infty \). But \( x \cdot v \) approaches \( 0 \) since \( x \) goes to \( 0 \) faster than \( \log(\sin x) \) diverges.

At \( x = \pi \):

Similarly, \( \sin \pi = 0 \) leads to \( \log(\sin \pi) \to -\infty \), but again, \( \pi \cdot v \) approaches \( 0 \) as \( \sin \pi \) vanishes. Therefore, the boundary contributions are both \( 0 \).

Step 5: Focus on the Remaining Integral

Now we must evaluate:

∫_{0}^{\pi} v \, dx

This part involves the integral of \( \log(\sin x) \), which can be known from integral tables or advanced calculus techniques. The value of this integral is:

∫_{0}^{\pi} \log(\sin x) \, dx = -\pi \log(2)

Step 6: Final Calculation

Putting this all together, we have:

∫_{0}^{\pi} x \log(\sin x) \, dx = 0 - \left(-\pi \log(2)\right) = \pi \log(2)

Conclusion

Therefore, the integral of \( x \log(\sin x) \) from \( 0 \) to \( \pi \) is:

∫_{0}^{\pi} x \log(\sin x) \, dx = \pi \log(2)

This result showcases the beauty of integration by parts and the fascinating properties of logarithmic functions combined with trigonometric identities. If you have any further questions or need clarification on any steps, feel free to ask!