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 $$$\int_{\frac{1}{3}}^{\frac{3}{4}}\left( 6 x \left(1 - x\right) \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{6 x \left(1 - x\right) d x}=x^{2} \left(3 - 2 x\right)$$$ (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.

$$$\left(x^{2} \left(3 - 2 x\right)\right)|_{\left(x=\frac{3}{4}\right)}=\frac{27}{32}$$$

$$$\left(x^{2} \left(3 - 2 x\right)\right)|_{\left(x=\frac{1}{3}\right)}=\frac{7}{27}$$$

$$$\int_{\frac{1}{3}}^{\frac{3}{4}}\left( 6 x \left(1 - x\right) \right)dx=\left(x^{2} \left(3 - 2 x\right)\right)|_{\left(x=\frac{3}{4}\right)}-\left(x^{2} \left(3 - 2 x\right)\right)|_{\left(x=\frac{1}{3}\right)}=\frac{505}{864}$$$

Answer: $$$\int_{\frac{1}{3}}^{\frac{3}{4}}\left( 6 x \left(1 - x\right) \right)dx=\frac{505}{864}\approx 0.584490740740741$$$

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