What is the integral of (z+5)/(z(z^2+49)) ?

The integral of (z+5)/(z(z^2+49)) is 1/7 arctan(z/7)+5(1/49 ln|z|-1/98 ln|z^2+49|)+C

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integral of (z+5)/(z(z^2+49))

\int \frac{z + 5}{z ( z ^{2} + 49 )} d z

\frac{1}{7} arctan (\frac{z}{7}) + 5 (\frac{1}{49} ln ∣ z ∣ - \frac{1}{98} ln z^{2} + 49) + C

Solution steps

\int \frac{z + 5}{z ( z ^{2} + 49 )} d z

Expand

\frac{z + 5}{z ( z ^{2} + 49 )} : \frac{z}{z ( z ^{2} + 49 )} + \frac{5}{z ( z ^{2} + 49 )}

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{z}{z ( z ^{2} + 49 )} d z + \int \frac{5}{z ( z ^{2} + 49 )} d z

\int \frac{z}{z ( z ^{2} + 49 )} d z = \frac{1}{7} arctan (\frac{z}{7})

\int \frac{5}{z ( z ^{2} + 49 )} d z = 5 (\frac{1}{49} ln ∣ z ∣ - \frac{1}{98} ln z^{2} + 49)

= \frac{1}{7} arctan (\frac{z}{7}) + 5 (\frac{1}{49} ln ∣ z ∣ - \frac{1}{98} ln z^{2} + 49)

Add a constant to the solution

= \frac{1}{7} arctan (\frac{z}{7}) + 5 (\frac{1}{49} ln ∣ z ∣ - \frac{1}{98} ln z^{2} + 49) + C

What is the integral of (z+5)/(z(z^2+49)) ?
The integral of (z+5)/(z(z^2+49)) is 1/7 arctan(z/7)+5(1/49 ln|z|-1/98 ln|z^2+49|)+C