Calcolatore di integrali definiti e impropri

Il calcolatore cercherà di valutare l'integrale definito (cioè con estremi), inclusi quelli impropri, mostrando i passaggi.

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Solution

Your input: calculate $$$\int_{\frac{1}{4}}^{\frac{1}{3}}\left( 3 - 6 x^{2} \right)dx$$$

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

$$$\left(x \left(3 - 2 x^{2}\right)\right)|_{\left(x=\frac{1}{3}\right)}=\frac{25}{27}$$$

$$$\left(x \left(3 - 2 x^{2}\right)\right)|_{\left(x=\frac{1}{4}\right)}=\frac{23}{32}$$$

$$$\int_{\frac{1}{4}}^{\frac{1}{3}}\left( 3 - 6 x^{2} \right)dx=\left(x \left(3 - 2 x^{2}\right)\right)|_{\left(x=\frac{1}{3}\right)}-\left(x \left(3 - 2 x^{2}\right)\right)|_{\left(x=\frac{1}{4}\right)}=\frac{179}{864}$$$

Answer: $$$\int_{\frac{1}{4}}^{\frac{1}{3}}\left( 3 - 6 x^{2} \right)dx=\frac{179}{864}\approx 0.207175925925926$$$

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Solution