How to prove correctness of a formula (differential calculus, integral)? How do I prove the correctness of the following formula relating to the fundamental theorem of calculus?
$$\int \! x\cos{3x} \, \mathrm{d}x = \frac{\cos{3x}}{9}+\frac{x\sin{3x}}{3}+C$$

Integration by parts would be of great help. Let $u = x$, and let $dv = \cos{3x} \:\rm{dx}$. Then you have $du = dx$ and $v = \frac{\sin{3x}}{3}$. Using the formula we have $$I = \int x \cos{3x} \ \text{dx} = uv - \int v \ \text{du} =  \frac{x \cdot\sin{3x}}{3} - \frac{1}{3}\int\sin{3x} \ \text{dx}$$
To see whether your answer is correct differentiate the Right Hand side. So you have $$ \frac{1}{9} \cdot -3\sin{3x} + \frac{1}{3} \sin{3x} + x \cos{3x} = x \cos{3x}$$
