Partielle Integration

$$\int u'v dx =u\cdot v-\int uv'dx$$

Merke dazu:

$$\int \frac{f'(x)}{f(x)}=\ln \vert f(x)\vert$$

Beispiel:

$$\int \underbrace{9x^{2}}_{u'}\underbrace{\ln\vert x}_{v} \qquad \... ...ll} u'=9x^{2} & v=\ln \vert x\vert\\ u=3x^{3} & v'=\frac{1}{x} \\ \end{array}$$

$$=3x^{3}\ln\vert x\vert-\int3x^{3}\cdot \frac{1}{x}dx = 3x^{3}\ln\vert x\vert-\int 3x^{2}dx$$

$$=3x^{3}\ln\vert x\vert-x^{3}+\underbrace{C}_{\textrm{\tiny {nicht vergessen!}}}=2x^{3}\ln\vert x\vert+C$$