Solve ∫ (from 0 to w) of 1/2uddu

$$\int _ { 0 } ^ { w } \frac { 1 } { 2 } u d d u$$

Evaluate

$$\frac{dw^{2}}{4}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{ud}{2}\mathrm{d}u$$

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

$$\frac{d}{2}\int u\mathrm{d}u$$

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}.

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

Simplify.

$$\frac{du^{2}}{4}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{4}dw^{2}-\frac{1}{4}d\times 0^{2}$$

Simplify.

$$\frac{dw^{2}}{4}$$

Differentiate w.r.t. d

$$\frac{w^{2}}{4}$$