∫[tex] \frac{x ^{3} }{ x^{2} +1} [/tex] dx
Facem substitutia t=x²+1
2x dx =dt
∫(t) =∫[tex] \frac{t-1}{2t} [/tex] dt = [tex] \frac{1}{2} [/tex] *∫[tex] \frac{t-1}{t} [/tex]dt
∫(t)=[tex] \frac{1}{2} [/tex]*(∫1 dt -∫[tex] \frac{1}{t} [/tex] dt )
∫(t)=[tex] \frac{1}{2} [/tex] * (t- ln|t|)
Revenim la integrala initiala si avem
∫(x)=\frac{1}{2}\left(x^2+1-\ln \left|x^2+1\right|\right)
Adaugam multimea de constante
∫(x)=\frac{1}{2}\left(x^2+1-\ln \left|x^2+1\right|\right)+C
