What is the integral of z(a+bz^3)^2 ?

The integral of z(a+bz^3)^2 is (a^2z^2)/2+(2abz^5)/5+(b^2z^8)/8+C

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integral of z(a+bz^3)^2

\int z (a + b z^{3})^{2} d z

\frac{a ^{2} z ^{2}}{2} + \frac{2 ab z ^{5}}{5} + \frac{b ^{2} z ^{8}}{8} + C

Solution steps

\int z (a + b z^{3})^{2} d z

Expand

z (a + b z^{3})^{2} : a^{2} z + 2 ab z^{4} + b^{2} z^{7}

= \int a^{2} z + 2 ab z^{4} + b^{2} z^{7} 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 a^{2} z d z + \int 2 ab z^{4} d z + \int b^{2} z^{7} d z

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

\int 2 ab z^{4} d z = \frac{2 ab z ^{5}}{5}

\int b^{2} z^{7} d z = \frac{b ^{2} z ^{8}}{8}

= \frac{a ^{2} z ^{2}}{2} + \frac{2 ab z ^{5}}{5} + \frac{b ^{2} z ^{8}}{8}

Add a constant to the solution

= \frac{a ^{2} z ^{2}}{2} + \frac{2 ab z ^{5}}{5} + \frac{b ^{2} z ^{8}}{8} + C

x \to 0 lim (\frac{1}{x cot ( x )})

y = x (4.2)

\int \frac{1}{n} sin (n t) d t

\int \frac{1}{2 + cos ( x )} d x

y^{^{'}} = x y^{\frac{1}{3}}

What is the integral of z(a+bz^3)^2 ?
The integral of z(a+bz^3)^2 is (a^2z^2)/2+(2abz^5)/5+(b^2z^8)/8+C