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calculate-x-3-4x-5-x-2-x-1-dx-

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Question Number 73477 by abdomathmax last updated on 13/Nov/19
calculate ∫ ((x^3 −4x+5)/(x^2 −x +1))dx
calculate∫x3−4x+5x2−x+1dx
Answered by MJS last updated on 13/Nov/19
∫((x^3 −4x+5)/(x^2 −x+1))dx= =∫xdx+∫dx−4∫((x−1)/(x^2 −x+1))dx ∫((x−1)/(x^2 −x+1))dx=(1/2)∫((2x−1)/(x^2 −x+1))dx−(1/2)∫(dx/(x^2 −x+1))= =(1/2)ln (x^2 −x+1) −((√3)/3)arctan (((2x−1)(√3))/3) ∫((x^3 −4x+5)/(x^2 −x+1))dx= =(1/2)x^2 +x−2ln (x^2 −x+1) +((4(√3))/3)arctan (((2x−1)(√3))/3) +C
∫x3−4x+5x2−x+1dx==∫xdx+∫dx−4∫x−1x2−x+1dx∫x−1x2−x+1dx=12∫2x−1x2−x+1dx−12∫dxx2−x+1==12ln(x2−x+1)−33arctan(2x−1)33∫x3−4x+5x2−x+1dx==12x2+x−2ln(x2−x+1)+433arctan(2x−1)33+C

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