Výpočet integrálu
Vypočítej hodnoty určitého integrálu pomocí metody per partes:
\large{\displaystyle\int\limits_1^{e}2x\ln{x}{d}x}
\large{\displaystyle\int\limits_1^{e}2x\ln{x}{d}x}=2\left(\left\lbrack\ln x\cdot\frac{x^2}{2}\right\rbrack_1^{e}-\frac12\left\lbrack\frac{x^2}{2}\right\rbrack_1^{e}\right)=
\large = 2 \left( \Large \frac{e^2}{2}\large -\Large \frac{1}{3}\large \cdot \Large \frac{e^2-1}{2}\large \right) = 2 \left( \Large \frac{e^2}{2}\large -\Large \frac{e^2-1}{6}\large \right) = \Large \frac{e^2+3}{4}\large
\large{\displaystyle\int\limits_1^{e}2x\ln{x}{d}x}=2\left(\left\lbrack\ln x\cdot\frac{x^2}{2}\right\rbrack_1^{e}-\frac12\left\lbrack\frac{x^2}{2}\right\rbrack_1^{e}\right)=
\large = 2 \left( \Large \frac{e^2}{2}\large -\Large \frac{1}{2}\large \cdot \Large \frac{e^2-1}{3}\large \right) = 2 \left( \Large \frac{e^2}{2}\large -\Large \frac{e^2-1}{6}\large \right) = \Large \frac{e^2+2}{3}\large
\large{\displaystyle\int\limits_1^{e}2x\ln{x}{d}x}=2\left(\left\lbrack\ln x\cdot\frac{x^2}{2}\right\rbrack_1^{e}-\frac12\left\lbrack\frac{x^2}{2}\right\rbrack_1^{e}\right)=
\large = 2 \left( \Large \frac{e^2}{2}\large -\Large \frac{1}{2}\large \cdot \Large \frac{e^2+1}{2}\large \right) = 2 \left( \Large \frac{e^2}{2}\large -\Large \frac{e^2+1}{4}\large \right) = \Large \frac{e^2-1}{2}\large
\large{\displaystyle\int\limits_1^{e}2x\ln{x}{d}x}=2\left(\left\lbrack\ln x\cdot\frac{x^2}{2}\right\rbrack_1^{e}-\frac12\left\lbrack\frac{x^2}{2}\right\rbrack_1^{e}\right)=
\large = 2 \left( \Large \frac{e^2}{2}\large -\Large \frac{1}{2}\large \cdot \Large \frac{e^2-1}{2}\large \right) = 2 \left( \Large \frac{e^2}{2}\large -\Large \frac{e^2-1}{4}\large \right) = \Large \frac{e^2+1}{2}\large