https://r2.erweima.ai/mathbot/image/1725495042562y07tbc15.jpg

Let's solve the integral 

\[
\int \sin x \cos x \ln(\sin x) \, dx
\]

step by step.

### Step 1: Simplify the Expression
Rewrite \(\sin x \cos x\) using the double-angle identity:

\[
\sin x \cos x = \frac{1}{2} \sin(2x).
\]

Thus, the integral becomes:

\[
\int \sin x \cos x \ln(\sin x) \, dx = \int \frac{1}{2} \sin(2x) \ln(\sin x) \, dx.
\]

Simplify further:

\[
= \frac{1}{2} \int \sin(2x) \ln(\sin x) \, dx.
\]

### Step 2: Use Substitution
Let \( u = \sin x \). Then, \( du = \cos x \, dx \), and hence \( dx = \frac{du}{\cos x} \).

We also know that:

\[
\cos x = \sqrt{1 - u^2}.
\]

Thus, the integral in terms of \( u \) is:

\[
\int \ln(u) \, du.
\]

Now, integrating this function:

\[
\int \ln u \, du = u\left(\ln u - 1\right) + C.
\]

Finally, substitute \( u = \sin x \):

\[
\int = \sin x (\ln (\sin x) - 1) + C.
\]

Solve the integral \( \int \sin x \cos x \ln(\sin x) \, dx \)

Learn how to solve the integral \( \int \sin x \cos x \ln(\sin x) \, dx \) using trigonometric identities and logarithmic integration. This problem covers key concepts such as the double-angle identity and logarithmic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation