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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Solution

Your input: calculate $$$\int_{0}^{2}\left( \ln{\left(x - 5 \right)} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\ln{\left(x - 5 \right)} d x}=\left(x - 5\right) \left(\ln{\left(x - 5 \right)} - 1\right)$$$ (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.

$$$\left(\left(x - 5\right) \left(\ln{\left(x - 5 \right)} - 1\right)\right)|_{\left(x=2\right)}=- 3 \ln{\left(3 \right)} + 3 - 3 i \pi$$$

$$$\left(\left(x - 5\right) \left(\ln{\left(x - 5 \right)} - 1\right)\right)|_{\left(x=0\right)}=- 5 \ln{\left(5 \right)} + 5 - 5 i \pi$$$

$$$\int_{0}^{2}\left( \ln{\left(x - 5 \right)} \right)dx=\left(\left(x - 5\right) \left(\ln{\left(x - 5 \right)} - 1\right)\right)|_{\left(x=2\right)}-\left(\left(x - 5\right) \left(\ln{\left(x - 5 \right)} - 1\right)\right)|_{\left(x=0\right)}=- 3 \ln{\left(3 \right)} - 2 + 5 \ln{\left(5 \right)} + 2 i \pi$$$

Answer: $$$\int_{0}^{2}\left( \ln{\left(x - 5 \right)} \right)dx=- 3 \ln{\left(3 \right)} - 2 + 5 \ln{\left(5 \right)} + 2 i \pi\approx 2.75135269616617 + 6.28318530717959 i$$$

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Solution