Solve ∫ (12u^2+4u)du

$$\int ( 12 u ^ { 2 } + 4 u ) d u$$

Evaluate

$$4u^{3}+2u^{2}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 12u^{2}\mathrm{d}u+\int 4u\mathrm{d}u$$

Factor out the constant in each of the terms.

$$12\int u^{2}\mathrm{d}u+4\int u\mathrm{d}u$$

Since \int u^{k}\mathrm{d}u=\frac{u^{k+1}}{k+1} for k\neq -1, replace \int u^{2}\mathrm{d}u with \frac{u^{3}}{3}. Multiply 12 times \frac{u^{3}}{3}.

$$4u^{3}+4\int u\mathrm{d}u$$

Since \int u^{k}\mathrm{d}u=\frac{u^{k+1}}{k+1} for k\neq -1, replace \int u\mathrm{d}u with \frac{u^{2}}{2}. Multiply 4 times \frac{u^{2}}{2}.

$$4u^{3}+2u^{2}$$

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

$$4u^{3}+2u^{2}+С$$

Differentiate w.r.t. u

$$4u\left(3u+1\right)$$