Υπολογιστής Ορισμένου και Ακατάλληλου Ολοκληρώματος

Υπολογίστε ορισμένα και ακατάλληλα ολοκληρώματα βήμα προς βήμα

Η αριθμομηχανή θα προσπαθήσει να υπολογίσει το ορισμένο (δηλ. με όρια) ολοκλήρωμα, συμπεριλαμβανομένου του ατελούς, παρουσιάζοντας τα βήματα.

Enter a function:

Integrate with respect to:

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

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Solution

Your input: calculate $$$\int_{0}^{2}\left( x^{3} - x^{2} - 2 x \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(x^{3} - x^{2} - 2 x\right)d x}=x^{2} \left(\frac{x^{2}}{4} - \frac{x}{3} - 1\right)$$$ (for steps, see indefinite integral calculator)

According to the Fundamental Theorem of Calculus, $$$\int_a^b F(x) dx=f(b)-f(a)$$$, so just evaluate the integral at the endpoints, and that's the answer.

$$$\left(x^{2} \left(\frac{x^{2}}{4} - \frac{x}{3} - 1\right)\right)|_{\left(x=2\right)}=- \frac{8}{3}$$$

$$$\left(x^{2} \left(\frac{x^{2}}{4} - \frac{x}{3} - 1\right)\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{2}\left( x^{3} - x^{2} - 2 x \right)dx=\left(x^{2} \left(\frac{x^{2}}{4} - \frac{x}{3} - 1\right)\right)|_{\left(x=2\right)}-\left(x^{2} \left(\frac{x^{2}}{4} - \frac{x}{3} - 1\right)\right)|_{\left(x=0\right)}=- \frac{8}{3}$$$

Answer: $$$\int_{0}^{2}\left( x^{3} - x^{2} - 2 x \right)dx=- \frac{8}{3}\approx -2.66666666666667$$$

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