Question

Cum se rezolva urmatoarea integrala: $\int\ {\frac{1}{x^3+x^5}} \, dx$

Answer 1

$\int \frac{1}{x^3+x^5}dx \\ =\int \frac{1}{x^3}+\frac{x}{x^2+1}-\frac{1}{x}dx \\ \int \frac{1}{x^3}dx+\int \frac{x}{x^2+1}dx-\int \frac{1}{x}dx \\ Calculam : \int \frac{1}{x^3}dx=-\frac{1}{2x^2} \\ \int \frac{x}{x^2+1}dx=\frac{1}{2}\ln \left|x^2+1\right| \\ \int \frac{x}{x^2+1}dx=\frac{1}{2}\ln \left|x^2+1\right| \\ Deci : \int \frac{1}{x^3}dx+\int \frac{x}{x^2+1}dx-\int \frac{1}{x}dx$$=-\frac{1}{2x^2}+\frac{1}{2}\ln \left|x^2+1\right|-\ln \left|x\right|=-\frac{1}{2x^2}+\frac{1}{2}\ln \left|x^2+1\right|-\ln \left|x\right|+C$