Solve ∫ -m^2+8m^3-8m^5dm

$$\int - m ^ { 2 } + 8 m ^ { 3 } - 8 m ^ { 5 } d m$$

Evaluate

$$-\frac{4m^{6}}{3}+2m^{4}-\frac{m^{3}}{3}+С$$

Solution Steps

Integrate the sum term by term.

$$\int -m^{2}\mathrm{d}m+\int 8m^{3}\mathrm{d}m+\int -8m^{5}\mathrm{d}m$$

Factor out the constant in each of the terms.

$$-\int m^{2}\mathrm{d}m+8\int m^{3}\mathrm{d}m-8\int m^{5}\mathrm{d}m$$

Since \int m^{k}\mathrm{d}m=\frac{m^{k+1}}{k+1} for k\neq -1, replace \int m^{2}\mathrm{d}m with \frac{m^{3}}{3}. Multiply -1 times \frac{m^{3}}{3}.

$$-\frac{m^{3}}{3}+8\int m^{3}\mathrm{d}m-8\int m^{5}\mathrm{d}m$$

Since \int m^{k}\mathrm{d}m=\frac{m^{k+1}}{k+1} for k\neq -1, replace \int m^{3}\mathrm{d}m with \frac{m^{4}}{4}. Multiply 8 times \frac{m^{4}}{4}.

$$-\frac{m^{3}}{3}+2m^{4}-8\int m^{5}\mathrm{d}m$$

Since \int m^{k}\mathrm{d}m=\frac{m^{k+1}}{k+1} for k\neq -1, replace \int m^{5}\mathrm{d}m with \frac{m^{6}}{6}. Multiply -8 times \frac{m^{6}}{6}.

$$-\frac{m^{3}}{3}+2m^{4}-\frac{4m^{6}}{3}$$

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

$$-\frac{m^{3}}{3}+2m^{4}-\frac{4m^{6}}{3}+С$$

Differentiate w.r.t. m

$$m^{2}\left(-8m^{3}+8m-1\right)$$