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 $$$\int_{0}^{\frac{\pi}{2}}\left( \frac{x \sin{\left(3 \right)}}{2} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\frac{x \sin{\left(3 \right)}}{2} d x}=\frac{x^{2} \sin{\left(3 \right)}}{4}$$$ (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.

$$$\left(\frac{x^{2} \sin{\left(3 \right)}}{4}\right)|_{\left(x=\frac{\pi}{2}\right)}=\frac{\pi^{2} \sin{\left(3 \right)}}{16}$$$

$$$\left(\frac{x^{2} \sin{\left(3 \right)}}{4}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{\frac{\pi}{2}}\left( \frac{x \sin{\left(3 \right)}}{2} \right)dx=\left(\frac{x^{2} \sin{\left(3 \right)}}{4}\right)|_{\left(x=\frac{\pi}{2}\right)}-\left(\frac{x^{2} \sin{\left(3 \right)}}{4}\right)|_{\left(x=0\right)}=\frac{\pi^{2} \sin{\left(3 \right)}}{16}$$$

Answer: $$$\int_{0}^{\frac{\pi}{2}}\left( \frac{x \sin{\left(3 \right)}}{2} \right)dx=\frac{\pi^{2} \sin{\left(3 \right)}}{16}\approx 0.0870499157893395$$$

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Kalkylator för bestämda och oegentliga integraler

Beräkna bestämda och oegentliga integraler steg för steg

Solution