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

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If you need `oo`, type inf.

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Solution

Your input: calculate $$$\int_{0}^{3}\left( \frac{x \left(x - 9\right) \left(x - 6\right)}{9} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\frac{x \left(x - 9\right) \left(x - 6\right)}{9} d x}=\frac{x^{2} \left(x^{2} - 20 x + 108\right)}{36}$$$ (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.

$$$\left(\frac{x^{2} \left(x^{2} - 20 x + 108\right)}{36}\right)|_{\left(x=3\right)}=\frac{57}{4}$$$

$$$\left(\frac{x^{2} \left(x^{2} - 20 x + 108\right)}{36}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{3}\left( \frac{x \left(x - 9\right) \left(x - 6\right)}{9} \right)dx=\left(\frac{x^{2} \left(x^{2} - 20 x + 108\right)}{36}\right)|_{\left(x=3\right)}-\left(\frac{x^{2} \left(x^{2} - 20 x + 108\right)}{36}\right)|_{\left(x=0\right)}=\frac{57}{4}$$$

Answer: $$$\int_{0}^{3}\left( \frac{x \left(x - 9\right) \left(x - 6\right)}{9} \right)dx=\frac{57}{4}=14.25$$$

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Solution