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 $$$\int_{0}^{1}\left( t^{2} x - t^{2} y \right)dx$$$

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

$$$\left(\frac{t^{2} x \left(x - 2 y\right)}{2}\right)|_{\left(x=1\right)}=\frac{t^{2} \left(1 - 2 y\right)}{2}$$$

$$$\left(\frac{t^{2} x \left(x - 2 y\right)}{2}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{1}\left( t^{2} x - t^{2} y \right)dx=\left(\frac{t^{2} x \left(x - 2 y\right)}{2}\right)|_{\left(x=1\right)}-\left(\frac{t^{2} x \left(x - 2 y\right)}{2}\right)|_{\left(x=0\right)}=\frac{t^{2} \left(1 - 2 y\right)}{2}$$$

Answer: $$$\int_{0}^{1}\left( t^{2} x - t^{2} y \right)dx=\frac{t^{2} \left(1 - 2 y\right)}{2}$$$

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Solution