Quotient de polynômes

Exemple 1

∫((2(x^2)+x+1)/((x^2)-3x+4))dx
=∫((2(x^2)-6x+8+7x-7)/((x^2)-3x+4))dx
=∫(2+((7x-7)/((x^2)-3x+4)))dx
=2x+(7/2)∫((2x-3+1)/((x^2)-3x+4))dx
=2x+(7/2)∫((2x-3)/((x^2)-3x+4))dx+(7/2)∫(1/((x^2)-3x+4))dx
=2x+(7/2)ln|(x^2)-3x+4|+(7/2)∫(1/((x^2)-3x+(9/4)+(7/4)))dx
=2x+(7/2)ln|(x^2)-3x+4|+(7/2)∫(1/((((x-(3/2)))^2)+(7/4)))dx
=2x+(7/2)ln|(x^2)-3x+4|+(7/2). (4/7)∫(1/(((((x-(3/2)))^2)/(7/4))+1))dx
=2x+(7/2)ln|(x^2)-3x+4| +2∫(1/((((((2x-3)/(Sqrt[7]))))^2)+1))dx
=2x+(7/2)ln|(x^2)-3x+4| +2.((Sqrt[7])/2)∫((2/(Sqrt[7]))/((((((2x-3)/(Sqrt[7]))))^2)+1))dx
=2x+(7/2)ln|(x^2)-3x+4| + (Sqrt[7])Arctg(((2x-3)/(Sqrt[7])))+k

Exemple  2

∫(((x^2)+2x-5)/((x^2)-3x+2))dx
=∫(((x^2)-3x+2+5x-7)/((x^2)-3x+2))dx
=∫(1+((5x-7)/((x^2)-3x+2)))dx
=x+(1/2)∫((10x-15+1)/((x^2)-3x+2))dx
=x+(1/2)∫((10x-15)/((x^2)-3x+2))dx+(1/2)∫(1/((x^2)-3x+2))dx
=x+(5/2)ln|(x^2)-3x+2|+(1/2)∫(1/((x-1)(x-2)))dx
=x+(5/2)ln|(x^2)-3x+2|+(1/2)∫((1/(x-2))-(1/(x-1)))dx
=x+(5/2)ln|(x^2)-3x+2|+(1/2)ln|((x-2)/(x-1))|+k

oubien
= ∫(1+((5x-7)/((x^2)-3x+2)))dx
= x + ∫((3x-3+2x-4)/((x-2)(x-1)))dx
= x +∫((3/(x-2))+(2/(x-1)))dx
=x+3ln|x-2|+2ln|x-1|+k

Exemple  3

∫(((x^2)+5x-2)/(2x-3))dx
=(1/2)∫((2(x^2)-3x+13x-4)/(2x-3))dx
=(1/2)∫(x+((13x-4)/(2x-3)))dx
=((x^2)/4)+(1/2)∫((13x-4)/(2x-3))dx
=((x^2)/4)+(1/4)∫((26x-39+31)/(2x-3))dx

=((x^2)/4)+(13/4)∫((2x-3)/(2x-3))dx+(1/4)∫(31/(2x-3))dx
=((x^2)/4)+((13x)/4)+(1/4)∫(31/(2x-3))dx
=((x^2)/4)+((13x)/4)+(31/8)∫(2/(2x-3))dx
=((x^2)/4)+((13x)/4)+(31/8)ln|2x-3|+k

Exercices

Calculer

EnoncePrimitives[(x+3)/(x^2+6),x];
EnoncePrimitives[(x+1)/(x^2-x-6),x];
EnoncePrimitives[(2x-1)/(x^2+3),x];
EnoncePrimitives[(2x^2+x-1)/(x+2),x];
EnoncePrimitives[(x-3)/(x^2+2x),x];

Solutions

CalculPrimitives[(x+3)/(x^2+6),x];
CalculPrimitives[(x+1)/(x^2-x-6),x];
CalculPrimitives[(2x-1)/(x^2+3),x];
CalculPrimitives[(2x^2+x-1)/(x+2),x];
CalculPrimitives[(x-3)/(x^2+2x),x];