Solve ∫ (from a to 2) of f(x)

$$\int _ { a } ^ { 2 } f ( x )$$

Evaluate

$$-\frac{fa^{2}}{2}+2f$$

Solution Steps

Evaluate the indefinite integral first.

$$\int fx\mathrm{d}x$$

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

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

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

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

Simplify.

$$\frac{fx^{2}}{2}$$

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.

$$\frac{1}{2}f\times 2^{2}-\frac{1}{2}fa^{2}$$

Simplify.

$$2f-\frac{fa^{2}}{2}$$

Differentiate w.r.t. f

$$-\frac{a^{2}}{2}+2$$