Solve ∫ ugdu

$$\int u g d u$$

Evaluate

$$\frac{gu^{2}}{2}+С$$

Solution Steps

Factor out the constant using \int af\left(u\right)\mathrm{d}u=a\int f\left(u\right)\mathrm{d}u.

$$g\int u\mathrm{d}u$$

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}.

$$g\times \frac{u^{2}}{2}$$

Simplify.

$$\frac{gu^{2}}{2}$$

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.

$$\frac{gu^{2}}{2}+С$$

Differentiate w.r.t. u

$$gu$$