What is the integral of (z^4+10z^2+25)/(z^3+5z) ?

The integral of (z^4+10z^2+25)/(z^3+5z) is (z^2)/2+5ln|z|+C

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integral of (z^4+10z^2+25)/(z^3+5z)

\int \frac{z ^{4} + 10 z ^{2} + 25}{z ^{3} + 5 z} d z

\frac{z ^{2}}{2} + 5 ln ∣ z ∣ + C

Solution steps

\int \frac{z ^{4} + 10 z ^{2} + 25}{z ^{3} + 5 z} d z

Simplify

\frac{z ^{4} + 10 z ^{2} + 25}{z ^{3} + 5 z} : \frac{z ^{2} + 5}{z}

= \int \frac{z ^{2} + 5}{z} d z

Expand

\frac{z ^{2} + 5}{z} : z + \frac{5}{z}

= \int z + \frac{5}{z} d z

Apply the Sum Rule:

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

= \int z d z + \int \frac{5}{z} d z

\int z d z = \frac{z ^{2}}{2}

\int \frac{5}{z} d z = 5 ln ∣ z ∣

= \frac{z ^{2}}{2} + 5 ln ∣ z ∣

Add a constant to the solution

= \frac{z ^{2}}{2} + 5 ln ∣ z ∣ + C

What is the integral of (z^4+10z^2+25)/(z^3+5z) ?
The integral of (z^4+10z^2+25)/(z^3+5z) is (z^2)/2+5ln|z|+C