Calcolatore di integrali definiti e impropri

Il calcolatore cercherà di valutare l'integrale definito (cioè con estremi), inclusi quelli impropri, mostrando i passaggi.

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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 $$$\int_{1}^{e}\left( \ln{\left(x \right)} \right)dx$$$

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

$$$\left(x \left(\ln{\left(x \right)} - 1\right)\right)|_{\left(x=e\right)}=0$$$

$$$\left(x \left(\ln{\left(x \right)} - 1\right)\right)|_{\left(x=1\right)}=-1$$$

$$$\int_{1}^{e}\left( \ln{\left(x \right)} \right)dx=\left(x \left(\ln{\left(x \right)} - 1\right)\right)|_{\left(x=e\right)}-\left(x \left(\ln{\left(x \right)} - 1\right)\right)|_{\left(x=1\right)}=1$$$

Answer: $$$\int_{1}^{e}\left( \ln{\left(x \right)} \right)dx=1$$$

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Solution