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 $$$\int_{0}^{x}\left( 2 \sin{\left(2 t \right)} \right)dt$$$

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

$$$\left(- \cos{\left(2 t \right)}\right)|_{\left(t=x\right)}=- \cos{\left(2 x \right)}$$$

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

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

Answer: $$$\int_{0}^{x}\left( 2 \sin{\left(2 t \right)} \right)dt=1 - \cos{\left(2 x \right)}$$$

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