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:

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 $$$\int_{0}^{2}\left( x e^{- x} \right)dx$$$

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

$$$\left(\left(- x - 1\right) e^{- x}\right)|_{\left(x=2\right)}=- \frac{3}{e^{2}}$$$

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

$$$\int_{0}^{2}\left( x e^{- x} \right)dx=\left(\left(- x - 1\right) e^{- x}\right)|_{\left(x=2\right)}-\left(\left(- x - 1\right) e^{- x}\right)|_{\left(x=0\right)}=1 - \frac{3}{e^{2}}$$$

Answer: $$$\int_{0}^{2}\left( x e^{- x} \right)dx=1 - \frac{3}{e^{2}}\approx 0.593994150290162$$$

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Solution