정적분 및 가적분 계산기

이 계산기는 단계별 풀이를 보여 주면서 상한과 하한이 있는 정적분(진정적분 포함)을 계산하려고 시도합니다.

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Solution

Your input: calculate $$$\int_{0}^{1}\left( e^{- x^{2}} \right)dx$$$

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.

$$$\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)|_{\left(x=1\right)}=\frac{\sqrt{\pi} \operatorname{erf}{\left(1 \right)}}{2}$$$

$$$\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{1}\left( e^{- x^{2}} \right)dx=\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)|_{\left(x=1\right)}-\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)|_{\left(x=0\right)}=\frac{\sqrt{\pi} \operatorname{erf}{\left(1 \right)}}{2}$$$

Answer: $$$\int_{0}^{1}\left( e^{- x^{2}} \right)dx=\frac{\sqrt{\pi} \operatorname{erf}{\left(1 \right)}}{2}\approx 0.746824132812427$$$

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정적분 및 가적분 계산기

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Solution