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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Solution

Your input: calculate $$$\int_{\frac{1}{2}}^{1}\left( \sin{\left(x \right)} - \frac{1}{x} \right)dx$$$

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

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

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

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

Answer: $$$\int_{\frac{1}{2}}^{1}\left( \sin{\left(x \right)} - \frac{1}{x} \right)dx=- \ln{\left(2 \right)} - \cos{\left(1 \right)} + \cos{\left(\frac{1}{2} \right)}\approx -0.355866924537712$$$

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Solution