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 $$$\int_{0}^{\pi}\left( x \sin{\left(x \right)} \right)dx$$$

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

$$$\left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right)|_{\left(x=\pi\right)}=\pi$$$

$$$\left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{\pi}\left( x \sin{\left(x \right)} \right)dx=\left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right)|_{\left(x=\pi\right)}-\left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right)|_{\left(x=0\right)}=\pi$$$

Answer: $$$\int_{0}^{\pi}\left( x \sin{\left(x \right)} \right)dx=\pi\approx 3.14159265358979$$$

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Kalkylator för bestämda och oegentliga integraler

Beräkna bestämda och oegentliga integraler steg för steg

Solution