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Calculadora de integrales definidas e impropias
Calcular integrales definidas e impropias paso a paso
La calculadora intentará evaluar la integral definida (es decir, con límites de integración), incluyendo las impropias, mostrando los pasos.
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: $$$\int_{2}^{1}\left( \frac{\ln{\left(x \right)}}{x^{2}} \right)dx=- \frac{1}{2} + \frac{\ln{\left(2 \right)}}{2}\approx -0.153426409720027$$$