Solve ∫ (from 4 to 25) of f(u)du

$$\int _ { 4 } ^ { 25 } f ( u ) d u$$

Evaluate

$$\frac{609f}{2}$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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}.

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

Simplify.

$$\frac{fu^{2}}{2}$$

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.

$$\frac{1}{2}f\times 25^{2}-\frac{1}{2}f\times 4^{2}$$

Simplify.

$$\frac{609f}{2}$$

Differentiate w.r.t. f

$$\frac{609}{2} = 304\frac{1}{2} = 304.5$$