Question

am nevoie mare de ajutor!!
cat face integrala din 1 supra x(x^2+2x+2)??

Answer 1

$\text{Vom descompune frac\c tia \^in frac\c tii simple:}\\\displaystyle\frac1{x(x^2+2x+2)}=\frac{A}{x}+\frac{Bx+C}{x^2+2x+2}\\\text{Aducem la acela\c si numitor \c si avem:}\\\displaystyle1=A(x^2+2x+2)+x(Bx+C)\\\displaystyle1=(A+B)x^2+(2A+C)x+2A\\\displaystyle A+B=0\\\displaystyle2A+C=0\\\displaystyle2A=1\\\displaystyle A=\frac12,B=-\frac12,C=-1\\\displaystyle\frac1{x(x^2+2x+2)}=\frac12\left(\frac1x-\frac{x+1}{x^2+2x+2}\right)$

$\displaystyle\int\frac12\left(\frac1x-\frac{x+1}{x^2+2x+2}\right)dx=\frac12\int\frac1xdx-\frac12\cdot\frac12\int\frac{2(x+1)}{x^2+2x+2}dx=\frac12\ln|x|-\frac14\ln|x^2+2x+2|+\matcal{C}$