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 $$$\int_{0}^{\pi}\left( \sin{\left(x^{2} \right)} \right)dx$$$

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

$$$\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=\pi\right)}=\frac{\sqrt{2} \sqrt{\pi} S\left(\sqrt{2} \sqrt{\pi}\right)}{2}$$$

$$$\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{\pi}\left( \sin{\left(x^{2} \right)} \right)dx=\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=\pi\right)}-\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=0\right)}=\frac{\sqrt{2} \sqrt{\pi} S\left(\sqrt{2} \sqrt{\pi}\right)}{2}$$$

Answer: $$$\int_{0}^{\pi}\left( \sin{\left(x^{2} \right)} \right)dx=\frac{\sqrt{2} \sqrt{\pi} S\left(\sqrt{2} \sqrt{\pi}\right)}{2}\approx 0.772651712690066$$$

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