What is the integral of (y^2+10y+25)/((y^2+25)^2) ?

The integral of (y^2+10y+25)/((y^2+25)^2) is 1/5 arctan(y/5)-5/(y^2+25)+C

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integral of (y^2+10y+25)/((y^2+25)^2)

\int \frac{y ^{2} + 10 y + 25}{( y ^{2} + 25 ) ^{2}} d y

\frac{1}{5} arctan (\frac{y}{5}) - \frac{5}{y ^{2} + 25} + C

Solution steps

\int \frac{y ^{2} + 10 y + 25}{( y ^{2} + 25 ) ^{2}} d y

Take the partial fraction of

\frac{y ^{2} + 10 y + 25}{( y ^{2} + 25 ) ^{2}} : \frac{1}{y ^{2} + 25} + \frac{10 y}{( y ^{2} + 25 ) ^{2}}

= \int \frac{1}{y ^{2} + 25} + \frac{10 y}{( y ^{2} + 25 ) ^{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} + 25} d y + \int \frac{10 y}{( y ^{2} + 25 ) ^{2}} d y

\int \frac{1}{y ^{2} + 25} d y = \frac{1}{5} arctan (\frac{y}{5})

\int \frac{10 y}{( y ^{2} + 25 ) ^{2}} d y = - \frac{5}{y ^{2} + 25}

= \frac{1}{5} arctan (\frac{y}{5}) - \frac{5}{y ^{2} + 25}

Add a constant to the solution

= \frac{1}{5} arctan (\frac{y}{5}) - \frac{5}{y ^{2} + 25} + C

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What is the integral of (y^2+10y+25)/((y^2+25)^2) ?
The integral of (y^2+10y+25)/((y^2+25)^2) is 1/5 arctan(y/5)-5/(y^2+25)+C