Subject 1.

Prof. Mihaela Berindeanu, Bucharest

| Details of solution | Associated marking scheme |
| :---: | :---: |
| $\int_{\pi}^{2 \pi} f(x) \mathrm{d} x=\int_{\pi}^{2 \pi} \frac{1}{x^{2}} \sin x \mathrm{~d} x+6 \int_{\pi}^{2 \pi} \frac{1}{x^{4}} \sin x \mathrm{~d} x$ | 1 point |
| $=\int_{\pi}^{2 \pi} \frac{\sin x}{x^{2}} \mathrm{~d} x-2 \int_{\pi}^{2 \pi}\left(\frac{1}{x^{3}}\right)^{\prime} \sin x \mathrm{~d} x=\int_{\pi}^{2 \pi} \frac{\sin x}{x^{2}} \mathrm{~d} x-\left.\frac{2 \sin x}{x^{3}} \sin x\right\|_{\pi} ^{2 \pi}+2 \int_{\pi}^{2 \pi} \frac{\cos x}{x^{3}} \mathrm{~d} x=$ | 3 points |
| $=\int_{\pi}^{2 \pi} \frac{\sin x}{x^{2}} \mathrm{~d} x-\int_{\pi}^{2 \pi}\left(\frac{1}{x^{2}}\right)^{\prime} \cos x \mathrm{~d} x=\int_{\pi}^{2 \pi} \frac{\sin x}{x^{2}} \mathrm{~d} x-\left.\frac{\cos x}{x^{2}}\right\|_{\pi} ^{2 \pi}-\int_{\pi}^{2 \pi} \frac{\sin x}{x^{2}} \mathrm{~d} x=-\left.\frac{\cos x}{x^{2}}\right\|_{\pi} ^{2 \pi}=$ | 3 points |
| $=-\frac{5}{4 \pi^{2}}$ |  |
Problem 1. Calculate

$$
\int_{\pi}^{2 \pi} \frac{x^{2}+6}{x^{4}} \sin x \mathrm{~d} x
$$