What is the integral of (y^2+12y+36)/((y^2+36)^2) ?

The integral of (y^2+12y+36)/((y^2+36)^2) is 1/6 arctan(y/6)-6/(y^2+36)+C

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integral of (y^2+12y+36)/((y^2+36)^2)

\int \frac{y ^{2} + 12 y + 36}{( y ^{2} + 36 ) ^{2}} d y

\frac{1}{6} arctan (\frac{y}{6}) - \frac{6}{y ^{2} + 36} + C

Solution steps

\int \frac{y ^{2} + 12 y + 36}{( y ^{2} + 36 ) ^{2}} d y

Take the partial fraction of

\frac{y ^{2} + 12 y + 36}{( y ^{2} + 36 ) ^{2}} : \frac{1}{y ^{2} + 36} + \frac{12 y}{( y ^{2} + 36 ) ^{2}}

= \int \frac{1}{y ^{2} + 36} + \frac{12 y}{( y ^{2} + 36 ) ^{2}} 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{1}{y ^{2} + 36} d y + \int \frac{12 y}{( y ^{2} + 36 ) ^{2}} d y

\int \frac{1}{y ^{2} + 36} d y = \frac{1}{6} arctan (\frac{y}{6})

\int \frac{12 y}{( y ^{2} + 36 ) ^{2}} d y = - \frac{6}{y ^{2} + 36}

= \frac{1}{6} arctan (\frac{y}{6}) - \frac{6}{y ^{2} + 36}

Add a constant to the solution

= \frac{1}{6} arctan (\frac{y}{6}) - \frac{6}{y ^{2} + 36} + C

What is the integral of (y^2+12y+36)/((y^2+36)^2) ?
The integral of (y^2+12y+36)/((y^2+36)^2) is 1/6 arctan(y/6)-6/(y^2+36)+C