The conversation between human and AI assistant.
How do I evaluate the integral [math]\int\log x\sin x\,dx[/math]?
The derivative of [math]\log x[/math] is [math]\frac{1}{x}[/math], and the antiderivative of [math]\sin x[/math] is [-\cos x].
Excellent! Now we can use the formula for integration by parts: [math]\int\log x\sin x\,dx = -\log x\cos x + \int\frac{\cos x}{x}\,dx[/math].