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Belirli ve Uygunsuz İntegral Hesaplayıcı
Belirli ve uygunsuz integralleri adım adım hesaplayın
Hesaplayıcı, uygunsuz olanlar da dahil olmak üzere belirli (yani sınırları olan) integrali adımları göstererek hesaplamaya çalışacaktır.
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: $$$\int_{a f g n^{3} t^{2} v y e^{2}}^{t}\left( e^{- x^{2}} \right)dx=\frac{\sqrt{\pi} \operatorname{erf}{\left(t \right)}}{2} - \frac{\sqrt{\pi} \operatorname{erf}{\left(a f g n^{3} t^{2} v y e^{2} \right)}}{2}$$$