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 $$$\int_{e}^{0}\left( 2 - 3 x \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(2 - 3 x\right)d x}=\frac{x \left(4 - 3 x\right)}{2}$$$ (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.

$$$\left(\frac{x \left(4 - 3 x\right)}{2}\right)|_{\left(x=0\right)}=0$$$

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

$$$\int_{e}^{0}\left( 2 - 3 x \right)dx=\left(\frac{x \left(4 - 3 x\right)}{2}\right)|_{\left(x=0\right)}-\left(\frac{x \left(4 - 3 x\right)}{2}\right)|_{\left(x=e\right)}=- \frac{e \left(4 - 3 e\right)}{2}$$$

Answer: $$$\int_{e}^{0}\left( 2 - 3 x \right)dx=- \frac{e \left(4 - 3 e\right)}{2}\approx 5.64702049147788$$$

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Solution