What is the integral of ((z+2))/(z^2+3z-1) ?

The integral of ((z+2))/(z^2+3z-1) is

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integral of ((z+2))/(z^2+3z-1)

\int \frac{( z + 2 )}{z ^{2} + 3 z - 1} d z

\frac{1}{2} ln 4 z^{2} + 12 z - 4 - \frac{1}{2 13} ln \frac{1}{13} (2 z + 3) + 1 + \frac{1}{2 13} ln \frac{1}{13} (2 z + 3) - 1 + C

Solution steps

\int \frac{z + 2}{z ^{2} + 3 z - 1} d z

Expand

\frac{z + 2}{z ^{2} + 3 z - 1} : \frac{z}{z ^{2} + 3 z - 1} + \frac{2}{z ^{2} + 3 z - 1}

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 ^{2} + 3 z - 1} d z + \int \frac{2}{z ^{2} + 3 z - 1} d z

\int \frac{z}{z ^{2} + 3 z - 1} d z = 2 (\frac{1}{4} ln 4 z^{2} + 12 z - 4 + \frac{3}{4 13} (ln \frac{2 z + 3}{13} + 1 - ln \frac{2 z + 3}{13} - 1))

\int \frac{2}{z ^{2} + 3 z - 1} d z = - \frac{2}{13} (ln \frac{2 z + 3}{13} + 1 - ln \frac{2 z + 3}{13} - 1)

= 2 (\frac{1}{4} ln 4 z^{2} + 12 z - 4 + \frac{3}{4 13} (ln \frac{2 z + 3}{13} + 1 - ln \frac{2 z + 3}{13} - 1)) - \frac{2}{13} (ln \frac{2 z + 3}{13} + 1 - ln \frac{2 z + 3}{13} - 1)

Simplify

2 (\frac{1}{4} ln 4 z^{2} + 12 z - 4 + \frac{3}{4 13} (ln \frac{2 z + 3}{13} + 1 - ln \frac{2 z + 3}{13} - 1)) - \frac{2}{13} (ln \frac{2 z + 3}{13} + 1 - ln \frac{2 z + 3}{13} - 1) : \frac{1}{2} ln 4 z^{2} + 12 z - 4 - \frac{1}{2 13} ln \frac{1}{13} (2 z + 3) + 1 + \frac{1}{2 13} ln \frac{1}{13} (2 z + 3) - 1

= \frac{1}{2} ln 4 z^{2} + 12 z - 4 - \frac{1}{2 13} ln \frac{1}{13} (2 z + 3) + 1 + \frac{1}{2 13} ln \frac{1}{13} (2 z + 3) - 1

Add a constant to the solution

= \frac{1}{2} ln 4 z^{2} + 12 z - 4 - \frac{1}{2 13} ln \frac{1}{13} (2 z + 3) + 1 + \frac{1}{2 13} ln \frac{1}{13} (2 z + 3) - 1 + C

\int t sin (π t) d t

\frac{\partial}{\partial x} (5 \cdot (y + 3)^{3} \cdot (x - 4))

f (x) = (x^{2} + 2)^{2}

f (x) = (8 x^{3} + 3 x^{2})^{3}

\frac{\partial}{\partial x} (2 x y - 2 x^{2} y)

What is the integral of ((z+2))/(z^2+3z-1) ?
The integral of ((z+2))/(z^2+3z-1) is