Question:

Let $I = \int_{\pi/2}^{\pi} x^{\sin x} (1 + x \cos x \cdot \ln x + \sin x) dx$<br>Put $x^{\sin x} = t \Rightarrow \sin x \cdot \ln x = \ln t$<br>$\Rightarrow \left( \cos x \cdot \ln x + \frac{\sin x}{x} \right) dx = \frac{dt}{t}$<br>$\Rightarrow x^{\sin x} (x \cos x \ln x + \sin x) dx = x dt$<br>$\therefore \int x^{\sin x} (1 + x \cos x \cdot \ln x + \sin x) dx = \int (t dx + x dt) = \int d(xt)$<br>$\therefore I = \left[ x \cdot x^{\sin x} \right]_{\pi/2}^{\pi} = \pi - \frac{\pi^2}{4}$