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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Solution

Your input: calculate $$$\int_{0}^{a}\left( \sin^{2}{\left(\frac{\pi m x}{a} \right)} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\sin^{2}{\left(\frac{\pi m x}{a} \right)} d x}=- \frac{a \sin{\left(\frac{2 \pi m x}{a} \right)}}{4 \pi m} + \frac{x}{2}$$$ (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.

$$$\left(- \frac{a \sin{\left(\frac{2 \pi m x}{a} \right)}}{4 \pi m} + \frac{x}{2}\right)|_{\left(x=a\right)}=\frac{a}{2} - \frac{a \sin{\left(2 \pi m \right)}}{4 \pi m}$$$

$$$\left(- \frac{a \sin{\left(\frac{2 \pi m x}{a} \right)}}{4 \pi m} + \frac{x}{2}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{a}\left( \sin^{2}{\left(\frac{\pi m x}{a} \right)} \right)dx=\left(- \frac{a \sin{\left(\frac{2 \pi m x}{a} \right)}}{4 \pi m} + \frac{x}{2}\right)|_{\left(x=a\right)}-\left(- \frac{a \sin{\left(\frac{2 \pi m x}{a} \right)}}{4 \pi m} + \frac{x}{2}\right)|_{\left(x=0\right)}=\frac{a}{2} - \frac{a \sin{\left(2 \pi m \right)}}{4 \pi m}$$$

Answer: $$$\int_{0}^{a}\left( \sin^{2}{\left(\frac{\pi m x}{a} \right)} \right)dx=\frac{a}{2} - \frac{a \sin{\left(2 \pi m \right)}}{4 \pi m}$$$