Calculadora de Integrais Definidas e Impróprias

A calculadora tentará calcular a integral definida (isto é, com limites), inclusive no caso impróprio, com os passos mostrados.

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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 $$$\int_{2}^{1}\left( \frac{\ln{\left(x \right)}}{x^{2}} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\frac{\ln{\left(x \right)}}{x^{2}} d x}=\frac{- \ln{\left(x \right)} - 1}{x}$$$ (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.

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

$$$\left(\frac{- \ln{\left(x \right)} - 1}{x}\right)|_{\left(x=2\right)}=- \frac{1}{2} - \frac{\ln{\left(2 \right)}}{2}$$$

$$$\int_{2}^{1}\left( \frac{\ln{\left(x \right)}}{x^{2}} \right)dx=\left(\frac{- \ln{\left(x \right)} - 1}{x}\right)|_{\left(x=1\right)}-\left(\frac{- \ln{\left(x \right)} - 1}{x}\right)|_{\left(x=2\right)}=- \frac{1}{2} + \frac{\ln{\left(2 \right)}}{2}$$$

Answer: $$$\int_{2}^{1}\left( \frac{\ln{\left(x \right)}}{x^{2}} \right)dx=- \frac{1}{2} + \frac{\ln{\left(2 \right)}}{2}\approx -0.153426409720027$$$

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Calculadora de Integrais Definidas e Impróprias

Calcule integrais definidas e impróprias passo a passo

Solution