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:

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

Enter an upper limit:

If you need `oo`, type inf.

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Solution

Your input: calculate $$$\int_{-1}^{2}\left( 13 t^{4} \right)dt$$$

First, calculate the corresponding indefinite integral: $$$\int{13 t^{4} d t}=\frac{13 t^{5}}{5}$$$ (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.

$$$\left(\frac{13 t^{5}}{5}\right)|_{\left(t=2\right)}=\frac{416}{5}$$$

$$$\left(\frac{13 t^{5}}{5}\right)|_{\left(t=-1\right)}=- \frac{13}{5}$$$

$$$\int_{-1}^{2}\left( 13 t^{4} \right)dt=\left(\frac{13 t^{5}}{5}\right)|_{\left(t=2\right)}-\left(\frac{13 t^{5}}{5}\right)|_{\left(t=-1\right)}=\frac{429}{5}$$$

Answer: $$$\int_{-1}^{2}\left( 13 t^{4} \right)dt=\frac{429}{5}\approx 85.8$$$