What is the integral of (8z^2+2)/(2z^3+z) ?

The integral of (8z^2+2)/(2z^3+z) is ln|2z^2+1|+2ln|z|+C

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integral of (8z^2+2)/(2z^3+z)

\int \frac{8 z ^{2} + 2}{2 z ^{3} + z}

ln 2 z^{2} + 1 + 2 ln ∣ z ∣ + C

Solution steps

\int \frac{8 z ^{2} + 2}{2 z ^{3} + z} d z

Expand

\frac{8 z ^{2} + 2}{2 z ^{3} + z} : \frac{8 z ^{2}}{2 z ^{3} + z} + \frac{2}{2 z ^{3} + 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 \frac{8 z ^{2}}{2 z ^{3} + z} d z + \int \frac{2}{2 z ^{3} + z} d z

\int \frac{8 z ^{2}}{2 z ^{3} + z} d z = 2 ln 2 z^{2} + 1

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

= 2 ln 2 z^{2} + 1 + 2 ln ∣ z ∣ - ln 2 z^{2} + 1

Simplify

= ln 2 z^{2} + 1 + 2 ln ∣ z ∣

Add a constant to the solution

= ln 2 z^{2} + 1 + 2 ln ∣ z ∣ + C

\frac{d}{d x} (\frac{e ^{x} cos ( x )}{e ^{x} - 1})

\frac{d}{d t} (4 cos (2 t))

\frac{d y}{d x} = (\frac{y}{y - x})

\frac{d}{d x} (x (x + 4)^{\frac{1}{3}})

x \to 0 - lim (x^{3} + 2 x + 2)

What is the integral of (8z^2+2)/(2z^3+z) ?
The integral of (8z^2+2)/(2z^3+z) is ln|2z^2+1|+2ln|z|+C