F ( x ) = ∫ (from 1 to x) of (t^2-4)dt eching

$$F ( x ) = \int _ { 1 } ^ { x } ( t ^ { 2 } - 4 ) d t$$

Baholash

$$\frac{x^{3}-12x+11}{3}$$

Yechim qadamlari

Evaluate the indefinite integral first.

$$\int t^{2}-4\mathrm{d}t$$

Integrate the sum term by term.

$$\int t^{2}\mathrm{d}t+\int -4\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}.

$$\frac{t^{3}}{3}+\int -4\mathrm{d}t$$

Find the integral of -4 using the table of common integrals rule \int a\mathrm{d}t=at.

$$\frac{t^{3}}{3}-4t$$

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.

$$\frac{x^{3}}{3}-4x-\left(\frac{1^{3}}{3}-4\right)$$

Qisqartirish.

$$\frac{x^{3}}{3}-4x+\frac{11}{3}$$

x ga nisbatan hosilani topish

$$x^{2}-4$$