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.

Please write without any differentials such as `dx`, `dy` etc.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Solution

Your input: calculate $$$\int_{0}^{1}\left( 2^{x} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{2^{x} d x}=\frac{2^{x}}{\ln{\left(2 \right)}}$$$ (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.

$$$\left(\frac{2^{x}}{\ln{\left(2 \right)}}\right)|_{\left(x=1\right)}=\frac{2}{\ln{\left(2 \right)}}$$$

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

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

Answer: $$$\int_{0}^{1}\left( 2^{x} \right)dx=\frac{1}{\ln{\left(2 \right)}}\approx 1.44269504088896$$$

Please try a new game Rotatly