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定积分与广义积分计算器
逐步计算定积分和广义积分
该计算器将尝试计算定积分（即带上下限的积分），包括广义积分，并显示求解步骤。
Solution
Your input: calculate $$$\int_{a f g n^{3} t^{2} v y e^{2}}^{t}\left( e^{- x^{2}} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{e^{- x^{2}} d x}=\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}$$$ (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.
$$$\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)|_{\left(x=t\right)}=\frac{\sqrt{\pi} \operatorname{erf}{\left(t \right)}}{2}$$$
$$$\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)|_{\left(x=a f g n^{3} t^{2} v y e^{2}\right)}=\frac{\sqrt{\pi} \operatorname{erf}{\left(a f g n^{3} t^{2} v y e^{2} \right)}}{2}$$$
$$$\int_{a f g n^{3} t^{2} v y e^{2}}^{t}\left( e^{- x^{2}} \right)dx=\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)|_{\left(x=t\right)}-\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)|_{\left(x=a f g n^{3} t^{2} v y e^{2}\right)}=\frac{\sqrt{\pi} \operatorname{erf}{\left(t \right)}}{2} - \frac{\sqrt{\pi} \operatorname{erf}{\left(a f g n^{3} t^{2} v y e^{2} \right)}}{2}$$$
Answer: $$$\int_{a f g n^{3} t^{2} v y e^{2}}^{t}\left( e^{- x^{2}} \right)dx=\frac{\sqrt{\pi} \operatorname{erf}{\left(t \right)}}{2} - \frac{\sqrt{\pi} \operatorname{erf}{\left(a f g n^{3} t^{2} v y e^{2} \right)}}{2}$$$