定積分與廣義積分計算器

此計算器將嘗試計算定積分（即帶有上下限的積分），包含廣義積分，並顯示步驟。

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Integrate with respect to:

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

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If you need `oo`, type inf.

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Solution

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

First, calculate the corresponding indefinite integral: $$$\int{e^{- \sqrt{x}} d x}=2 \left(- \sqrt{x} - 1\right) e^{- \sqrt{x}}$$$ (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.

$$$\left(2 \left(- \sqrt{x} - 1\right) e^{- \sqrt{x}}\right)|_{\left(x=1\right)}=- \frac{4}{e}$$$

$$$\left(2 \left(- \sqrt{x} - 1\right) e^{- \sqrt{x}}\right)|_{\left(x=0\right)}=-2$$$

$$$\int_{0}^{1}\left( e^{- \sqrt{x}} \right)dx=\left(2 \left(- \sqrt{x} - 1\right) e^{- \sqrt{x}}\right)|_{\left(x=1\right)}-\left(2 \left(- \sqrt{x} - 1\right) e^{- \sqrt{x}}\right)|_{\left(x=0\right)}=2 - \frac{4}{e}$$$

Answer: $$$\int_{0}^{1}\left( e^{- \sqrt{x}} \right)dx=2 - \frac{4}{e}\approx 0.528482235314231$$$

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定積分與廣義積分計算器

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Solution