[tex] \int\limits} \, \frac{x^3-x+3}{x^2+x-2} dx [/tex]

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03-02-2017
Mathematics

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[tex] \int\limits} \, \frac{x^3-x+3}{x^2+x-2} dx [/tex]

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LammettHash LammettHash · Answer 1 · 2017-02-06T19:47:37+01:00

First, simplify the quotient:

[tex]\displaystyle\int\frac{x^3-x+3}{x^2+x-2}\,\mathrm dx=\int\left(x-1+\frac{2x+1}{x^2+x-2}\right)\,\mathrm dx[/tex]

The first two terms are trivial to deal with, while the last term can be taken care of with a substitution of [tex]y=x^2+x-2[/tex]. This gives [tex]\mathrm dy=(2x+1)\,\mathrm dx[/tex], and you have

[tex]\displaystyle\int (x-1)\,\mathrm dx+\int\frac{\mathrm du}u[/tex]
[tex]\dfrac12x^2-x+\ln|u|+C[/tex]
[tex]\dfrac12x^2-x+\ln|x^2+x-2|+C[/tex]