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.

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Solution

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

First, calculate the corresponding indefinite integral: $$$\int{\frac{1}{x} d x}=\ln{\left(\left|{x}\right| \right)}$$$ (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.

$$$\left(\ln{\left(\left|{x}\right| \right)}\right)|_{\left(x=e\right)}=1$$$

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

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

Answer: $$$\int_{1}^{e}\left( \frac{1}{x} \right)dx=1$$$

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