Definite and Improper Integral Calculator

The calculator will try to evaluate the definite (i.e. with bounds) integral, including improper, with steps shown.

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

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If you need `oo`, type inf.

Please write without any differentials such as `dx`, `dy` etc.

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Solution

Your input: calculate $$$\int_{-\infty}^{3}\left( \frac{\ln{\left(x \right)}}{x} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\frac{\ln{\left(x \right)}}{x} d x}=\frac{\ln{\left(\left|{x}\right| \right)}^{2}}{2}$$$ (for steps, see indefinite integral calculator)

Since there is infinity in the lower bound, this is an improper integral of type 1.

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 (infinite).

$$$\int_{-\infty}^{3}\left( \frac{\ln{\left(x \right)}}{x} \right)dx=\left(\frac{\ln{\left(\left|{x}\right| \right)}^{2}}{2}\right)|_{\left(x=3\right)}-\lim_{x \to -\infty}\left(\frac{\ln{\left(\left|{x}\right| \right)}^{2}}{2}\right)=-\infty$$$

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

Answer: the integral is divergent.

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Definite and Improper Integral Calculator

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Solution