定積分・広義積分計算機

この計算機は、定積分（すなわち上下限のある積分）を、広義積分も含めて、解法手順を示しながら計算を試みます。

Enter a function:

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 $$$\int_{0}^{1}\left( \frac{1}{x} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\frac{1}{x} d x}=\ln{\left(\left|{x}\right| \right)}$$$ (for steps, see indefinite integral calculator)

The interval of integration contains the point $$$0$$$, which is not in the domain of the integrand, so this is an improper integral of type 2.

To evaluate an integral over an interval, we use the Fundamental Theorem of Calculus. However, we need to use limit if an endpoint of the interval is special (is not in the domain of the function).

$$$\int_{0}^{1}\left( \frac{1}{x} \right)dx=\left(\ln{\left(\left|{x}\right| \right)}\right)|_{\left(x=1\right)}-\lim_{x \to 0}\left(\ln{\left(\left|{x}\right| \right)}\right)=\infty$$$

Since the value of the integral is not finite, then it is divergent.

Answer: the integral is divergent.

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定積分・広義積分計算機

定積分と広義積分を手順を追って計算する

Solution