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

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

$$$\left(2 \operatorname{atan}{\left(x \right)}\right)|_{\left(x=1\right)}=\frac{\pi}{2}$$$

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

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

Answer: $$$\int_{-1}^{1}\left( \frac{2}{x^{2} + 1} \right)dx=\pi\approx 3.14159265358979$$$

Kalkylator för bestämda och oegentliga integraler

Beräkna bestämda och oegentliga integraler steg för steg

Solution