What is the integral of (y^2-5y+6)/(y^2+4) ?

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

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integral of (y^2-5y+6)/(y^2+4)

\int \frac{y ^{2} - 5 y + 6}{y ^{2} + 4} d y

- \frac{5}{2} ln y^{2} + 4 + arctan (\frac{y}{2}) + y + C

Solution steps

\int \frac{y ^{2} - 5 y + 6}{y ^{2} + 4} d y

\frac{y ^{2} - 5 y + 6}{y ^{2} + 4} = \frac{- 5 y + 2}{y ^{2} + 4} + 1

= \int \frac{- 5 y + 2}{y ^{2} + 4} + 1 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 \frac{- 5 y + 2}{y ^{2} + 4} d y + \int 1 d y

\int \frac{- 5 y + 2}{y ^{2} + 4} d y = - \frac{5}{2} ln y^{2} + 4 + arctan (\frac{y}{2})

\int 1 d y = y

= - \frac{5}{2} ln y^{2} + 4 + arctan (\frac{y}{2}) + y

Add a constant to the solution

= - \frac{5}{2} ln y^{2} + 4 + arctan (\frac{y}{2}) + y + C

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