Solve ∫ (from - 1 to 2) of f(x) wrt x

$$\int _ { - 1 } ^ { 2 } f ( x ) d x$$

Evaluate

$$\frac{3f}{2}$$

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.

$$\frac{1}{2}f\times 2^{2}-\frac{1}{2}f\left(-1\right)^{2}$$

Simplify.

$$\frac{3f}{2}$$

Differentiate w.r.t. f

$$\frac{3}{2} = 1\frac{1}{2} = 1.5$$