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.

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If you need `oo`, type inf.

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Solution

Your input: calculate $$$\int_{\frac{\pi}{2}}^{\frac{\pi}{3}}\left( 36 \cos^{2}{\left(\theta \right)} \right)d\theta$$$

First, calculate the corresponding indefinite integral: $$$\int{36 \cos^{2}{\left(\theta \right)} d \theta}=18 \theta + 9 \sin{\left(2 \theta \right)}$$$ (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.

$$$\left(18 \theta + 9 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{3}\right)}=\frac{9 \sqrt{3}}{2} + 6 \pi$$$

$$$\left(18 \theta + 9 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{2}\right)}=9 \pi$$$

$$$\int_{\frac{\pi}{2}}^{\frac{\pi}{3}}\left( 36 \cos^{2}{\left(\theta \right)} \right)d\theta=\left(18 \theta + 9 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{3}\right)}-\left(18 \theta + 9 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{2}\right)}=- 3 \pi + \frac{9 \sqrt{3}}{2}$$$

Answer: $$$\int_{\frac{\pi}{2}}^{\frac{\pi}{3}}\left( 36 \cos^{2}{\left(\theta \right)} \right)d\theta=- 3 \pi + \frac{9 \sqrt{3}}{2}\approx -1.63054932670943$$$

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