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

The integral of z(1+2z)^2 is (z^2)/2+(4z^3)/3+z^4+C

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integral of z(1+2z)^2

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

\frac{z ^{2}}{2} + \frac{4 z ^{3}}{3} + z^{4} + C

Solution steps

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

Expand

z (1 + 2 z)^{2} : z + 4 z^{2} + 4 z^{3}

= \int z + 4 z^{2} + 4 z^{3} 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 4 z^{2} d z + \int 4 z^{3} d z

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

\int 4 z^{2} d z = \frac{4 z ^{3}}{3}

\int 4 z^{3} d z = z^{4}

= \frac{z ^{2}}{2} + \frac{4 z ^{3}}{3} + z^{4}

Add a constant to the solution

= \frac{z ^{2}}{2} + \frac{4 z ^{3}}{3} + z^{4} + C

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What is the integral of z(1+2z)^2 ?
The integral of z(1+2z)^2 is (z^2)/2+(4z^3)/3+z^4+C