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.

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If you need `oo`, type inf.

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Solution

Your input: calculate $$$\int_{4}^{8}\left( 8 x^{2} + 5 x - 9 \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(8 x^{2} + 5 x - 9\right)d x}=\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}$$$ (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.

$$$\left(\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}\right)|_{\left(x=8\right)}=\frac{4360}{3}$$$

$$$\left(\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}\right)|_{\left(x=4\right)}=\frac{524}{3}$$$

$$$\int_{4}^{8}\left( 8 x^{2} + 5 x - 9 \right)dx=\left(\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}\right)|_{\left(x=8\right)}-\left(\frac{x \left(16 x^{2} + 15 x - 54\right)}{6}\right)|_{\left(x=4\right)}=\frac{3836}{3}$$$

Answer: $$$\int_{4}^{8}\left( 8 x^{2} + 5 x - 9 \right)dx=\frac{3836}{3}\approx 1278.66666666667$$$