Integration by parts follows the equation
$$\int uv = u \int v - \int (u' \times \int v)$$
Example:
$$\int 6x \cos 2x \, dx$$
Let
$$u = 6x \qquad v = \cos 2x$$
Then
$$u' = 6 \qquad \int v = \frac{1}{2} \sin 2x$$
Using the equation:
$$6x \left( \frac{1}{2} \sin 2x \right) - \int 6 \left( \frac{1}{2} \sin 2x \right)$$
$$= 3x \sin 2x - \int 3 \sin 2x$$
$$= 3x \sin 2x - \left( -\frac{3}{2} \cos 2x \right)$$
$$= 3x \sin 2x + \frac{3}{2} \cos 2x$$