Kalkylator för bestämda och oegentliga integraler

Kalkylatorn försöker beräkna bestämda integraler (dvs. med integrationsgränser), inklusive oegentliga, och visar stegen.

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Solution

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

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

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

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

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

Answer: $$$\int_{1}^{2}\left( \frac{1}{x} \right)dx=\ln{\left(2 \right)}\approx 0.693147180559945$$$

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Kalkylator för bestämda och oegentliga integraler

Beräkna bestämda och oegentliga integraler steg för steg

Solution