What is the integral of (y^2)/(y^2+y-6) ?

The integral of (y^2)/(y^2+y-6) is y+4/5 ln|y-2|-9/5 ln|y+3|+C

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integral of (y^2)/(y^2+y-6)

\int \frac{y ^{2}}{y ^{2} + y - 6} d y

y + \frac{4}{5} ln ∣ y - 2 ∣ - \frac{9}{5} ln ∣ y + 3 ∣ + C

Solution steps

\int \frac{y ^{2}}{y ^{2} + y - 6} d y

Take the partial fraction of

\frac{y ^{2}}{y ^{2} + y - 6} : 1 + \frac{4}{5 ( y - 2 )} - \frac{9}{5 ( y + 3 )}

= \int 1 + \frac{4}{5 ( y - 2 )} - \frac{9}{5 ( y + 3 )} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d y + \int \frac{4}{5 ( y - 2 )} d y - \int \frac{9}{5 ( y + 3 )} d y

\int 1 d y = y

\int \frac{4}{5 ( y - 2 )} d y = \frac{4}{5} ln ∣ y - 2 ∣

\int \frac{9}{5 ( y + 3 )} d y = \frac{9}{5} ln ∣ y + 3 ∣

= y + \frac{4}{5} ln ∣ y - 2 ∣ - \frac{9}{5} ln ∣ y + 3 ∣

Add a constant to the solution

= y + \frac{4}{5} ln ∣ y - 2 ∣ - \frac{9}{5} ln ∣ y + 3 ∣ + C

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What is the integral of (y^2)/(y^2+y-6) ?
The integral of (y^2)/(y^2+y-6) is y+4/5 ln|y-2|-9/5 ln|y+3|+C