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 $$$\int_{2 \pi}^{0}\left( \sin{\left(\theta \right)} \right)d\theta$$$

First, calculate the corresponding indefinite integral: $$$\int{\sin{\left(\theta \right)} d \theta}=- \cos{\left(\theta \right)}$$$ (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.

$$$\left(- \cos{\left(\theta \right)}\right)|_{\left(\theta=0\right)}=-1$$$

$$$\left(- \cos{\left(\theta \right)}\right)|_{\left(\theta=2 \pi\right)}=-1$$$

$$$\int_{2 \pi}^{0}\left( \sin{\left(\theta \right)} \right)d\theta=\left(- \cos{\left(\theta \right)}\right)|_{\left(\theta=0\right)}-\left(- \cos{\left(\theta \right)}\right)|_{\left(\theta=2 \pi\right)}=0$$$

Answer: $$$\int_{2 \pi}^{0}\left( \sin{\left(\theta \right)} \right)d\theta=0$$$

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Kalkylator för bestämda och oegentliga integraler

Beräkna bestämda och oegentliga integraler steg för steg

Solution