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 $$$\int_{20}^{90}\left( 120040 - \frac{6002 x}{5} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(120040 - \frac{6002 x}{5}\right)d x}=\frac{3001 x \left(200 - x\right)}{5}$$$ (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.

$$$\left(\frac{3001 x \left(200 - x\right)}{5}\right)|_{\left(x=90\right)}=5941980$$$

$$$\left(\frac{3001 x \left(200 - x\right)}{5}\right)|_{\left(x=20\right)}=2160720$$$

$$$\int_{20}^{90}\left( 120040 - \frac{6002 x}{5} \right)dx=\left(\frac{3001 x \left(200 - x\right)}{5}\right)|_{\left(x=90\right)}-\left(\frac{3001 x \left(200 - x\right)}{5}\right)|_{\left(x=20\right)}=3781260$$$

Answer: $$$\int_{20}^{90}\left( 120040 - \frac{6002 x}{5} \right)dx=3781260$$$

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Solution