Integrace funkce ln(x) * x
Zintegruj funkci f(x)=\ln x \cdot x metodou per partes.
=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \int \frac{x^{2}}{x} \mathrm{~d} x=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \int x \mathrm{~d} x=
=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \cdot \frac{x}{2} \mathrm{~d} x+C=\frac{x^{2}}{2} \cdot \ln x-\frac{x}{4}+C
=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \int \frac{x^{2}}{x} \mathrm{~d} x=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \int x \mathrm{~d} x=
=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \cdot \frac{x^{2}}{2} \mathrm{~d} x+C=\frac{x^{2}}{2} \cdot \ln x-\frac{x^{2}}{4}+C
=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \int \frac{x^{2}}{x} \mathrm{~d} x=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \int x^{2} \mathrm{~d} x=
=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \cdot \frac{x^{3}}{3} \mathrm{~d} x+C=\frac{x^{2}}{2} \cdot \ln x-\frac{x^{3}}{6}+C
=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \int \frac{x^{2}}{x} \mathrm{~d} x=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \int x \mathrm{~d} x=
=\frac{x^{2}}{2} \cdot \ln x-\frac{1}{2} \cdot \frac{x^{2}}{3} \mathrm{~d} x+C=\frac{x^{2}}{2} \cdot \ln x-\frac{x^{2}}{6}+C