Mathematics: How to integrate with steps: (4x2-6)/(x+5)(x-2)(3x-1)

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How to integrate with steps: (4x2-6)/(x+5)(x-2)(3x-1)

Ответ:

aiffland113

09.03.2022

$\displaystyle\int\frac{4x^2-6}{(x+5)(x-2)(3x-1)}\,\mathrm dx$

You have a rational expression whose numerator's degree is smaller than the denominator's. This tells you you should consider a partial fraction decomposition. We want to rewrite the integrand in the form

$\dfrac{4x^2-6}{(x+5)(x-2)(3x-1)}=\dfrac a{x+5}+\dfrac b{x-2}+\dfrac c{3x-1}$

$\implies4x^2-6=a(x-2)(3x-1)+b(x+5)(3x-1)+c(x+5)(x-2)$

You can use the "cover-up" method here to easily solve for a,b,c . It involves fixing a value of to make 2 of the 3 terms on the right side disappear and leaving a simple algebraic equation to solve for the remaining one.