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 $$$\int_{a}^{b}\left( x^{- n} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{x^{- n} d x}=- \frac{x^{1 - n}}{n - 1}$$$ (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.

$$$\left(- \frac{x^{1 - n}}{n - 1}\right)|_{\left(x=b\right)}=- \frac{b^{1 - n}}{n - 1}$$$

$$$\left(- \frac{x^{1 - n}}{n - 1}\right)|_{\left(x=a\right)}=- \frac{a^{1 - n}}{n - 1}$$$

$$$\int_{a}^{b}\left( x^{- n} \right)dx=\left(- \frac{x^{1 - n}}{n - 1}\right)|_{\left(x=b\right)}-\left(- \frac{x^{1 - n}}{n - 1}\right)|_{\left(x=a\right)}=\frac{a^{1 - n}}{n - 1} - \frac{b^{1 - n}}{n - 1}$$$

Answer: $$$\int_{a}^{b}\left( x^{- n} \right)dx=\frac{a^{1 - n}}{n - 1} - \frac{b^{1 - n}}{n - 1}$$$

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Solution