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

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

$$$\left(- \frac{\cos{\left(2 x \right)}}{2}\right)|_{\left(x=\frac{\pi}{2}\right)}=\frac{1}{2}$$$

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

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

Answer: $$$\int_{0}^{\frac{\pi}{2}}\left( \sin{\left(2 x \right)} \right)dx=1$$$

Please try a new game Rotatly