定積分・広義積分計算機

この計算機は、定積分（すなわち上下限のある積分）を、広義積分も含めて、解法手順を示しながら計算を試みます。

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Integrate with respect to:

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

Please write without any differentials such as `dx`, `dy` etc.

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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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定積分・広義積分計算機

定積分と広義積分を手順を追って計算する

Solution