Solve ∫ (z^2+5)^3z wrt z

$$\int ( z ^ { 2 } + 5 ) ^ { 3 } z d z$$

Evaluate

$$\frac{z^{8}}{8}+\frac{5z^{6}}{2}+\frac{75z^{4}}{4}+\frac{125z^{2}}{2}+С$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(z^{2}+5\right)^{3}.

$$\int \left(\left(z^{2}\right)^{3}+15\left(z^{2}\right)^{2}+75z^{2}+125\right)z\mathrm{d}z$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\int \left(z^{6}+15\left(z^{2}\right)^{2}+75z^{2}+125\right)z\mathrm{d}z$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\int \left(z^{6}+15z^{4}+75z^{2}+125\right)z\mathrm{d}z$$

Use the distributive property to multiply z^{6}+15z^{4}+75z^{2}+125 by z.

$$\int z^{7}+15z^{5}+75z^{3}+125z\mathrm{d}z$$

Integrate the sum term by term.

$$\int z^{7}\mathrm{d}z+\int 15z^{5}\mathrm{d}z+\int 75z^{3}\mathrm{d}z+\int 125z\mathrm{d}z$$

Factor out the constant in each of the terms.

$$\int z^{7}\mathrm{d}z+15\int z^{5}\mathrm{d}z+75\int z^{3}\mathrm{d}z+125\int z\mathrm{d}z$$

Since \int z^{k}\mathrm{d}z=\frac{z^{k+1}}{k+1} for k\neq -1, replace \int z^{7}\mathrm{d}z with \frac{z^{8}}{8}.

$$\frac{z^{8}}{8}+15\int z^{5}\mathrm{d}z+75\int z^{3}\mathrm{d}z+125\int z\mathrm{d}z$$

Since \int z^{k}\mathrm{d}z=\frac{z^{k+1}}{k+1} for k\neq -1, replace \int z^{5}\mathrm{d}z with \frac{z^{6}}{6}. Multiply 15 times \frac{z^{6}}{6}.

$$\frac{z^{8}}{8}+\frac{5z^{6}}{2}+75\int z^{3}\mathrm{d}z+125\int z\mathrm{d}z$$

Since \int z^{k}\mathrm{d}z=\frac{z^{k+1}}{k+1} for k\neq -1, replace \int z^{3}\mathrm{d}z with \frac{z^{4}}{4}. Multiply 75 times \frac{z^{4}}{4}.

$$\frac{z^{8}}{8}+\frac{5z^{6}}{2}+\frac{75z^{4}}{4}+125\int z\mathrm{d}z$$

Since \int z^{k}\mathrm{d}z=\frac{z^{k+1}}{k+1} for k\neq -1, replace \int z\mathrm{d}z with \frac{z^{2}}{2}. Multiply 125 times \frac{z^{2}}{2}.

$$\frac{z^{8}}{8}+\frac{5z^{6}}{2}+\frac{75z^{4}}{4}+\frac{125z^{2}}{2}$$

Simplify.

$$\frac{125z^{2}}{2}+\frac{75z^{4}}{4}+\frac{5z^{6}}{2}+\frac{z^{8}}{8}$$

If F\left(z\right) is an antiderivative of f\left(z\right), then the set of all antiderivatives of f\left(z\right) is given by F\left(z\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{125z^{2}}{2}+\frac{75z^{4}}{4}+\frac{5z^{6}}{2}+\frac{z^{8}}{8}+С$$

Differentiate w.r.t. z

$$z\left(z^{2}+5\right)^{3}$$