Solve ∫ (5u^4+3u^2)du

$$\int ( 5 u ^ { 4 } + 3 u ^ { 2 } ) d u$$

Evaluate

$$u^{5}+u^{3}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 5u^{4}\mathrm{d}u+\int 3u^{2}\mathrm{d}u$$

Factor out the constant in each of the terms.

$$5\int u^{4}\mathrm{d}u+3\int u^{2}\mathrm{d}u$$

Since \int u^{k}\mathrm{d}u=\frac{u^{k+1}}{k+1} for k\neq -1, replace \int u^{4}\mathrm{d}u with \frac{u^{5}}{5}. Multiply 5 times \frac{u^{5}}{5}.

$$u^{5}+3\int u^{2}\mathrm{d}u$$

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 3 times \frac{u^{3}}{3}.

$$u^{5}+u^{3}$$

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.

$$u^{5}+u^{3}+С$$

Differentiate w.r.t. u

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