Question

Sa se calculeze primitivele funcției: f: D -> R
a) f (x)= 1 supra lui x - 1 supra lui x2 + 1 supra x3.

Answer 1

$\int \frac{1}{\frac{x-1}{\frac{x^2+1}{x^3}}}dx=\int \frac{x^2+1}{\left(x-1\right)x^3}dx=\int \:-\frac{1}{x^3}-\frac{1}{x^2}-\frac{2}{x}+\frac{2}{x-1}dx=-\int \frac{1}{x^3}dx-\int \frac{1}{x^2}dx-\int \frac{2}{x}dx+\int \frac{2}{x-1}dx=-\left(-\frac{1}{2x^2}\right)-\left(-\frac{1}{x}\right)-2\ln \left|x\right|+2\ln \left|x-1\right|=\frac{1}{2x^2}+\frac{1}{x}-2\ln \left|x\right|+2\ln \left|x-1\right|=\frac{1}{2x^2}+\frac{1}{x}-2\ln \left|x\right|+2\ln \left|x-1\right|+C$