Kalkylator för bestämda och oegentliga integraler

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

Enter a function:

Integrate with respect to:

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

Please write without any differentials such as `dx`, `dy` etc.

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Solution

Your input: calculate $$$\int_{\infty}^{0}\left( x e^{- x^{2}} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{x e^{- x^{2}} d x}=- \frac{e^{- x^{2}}}{2}$$$ (for steps, see indefinite integral calculator)

Since the upper bound is less than the lower, according to the property of integrals, we can swap them and change the sign:

$$$\int_{\infty}^{0}\left( x e^{- x^{2}} \right)dx=-\int_{0}^{\infty}\left( x e^{- x^{2}} \right)dx$$$

Since there is infinity in the upper bound, this is improper integral of type 1.

To evaluate an integral over an interval, we use the Fundamental Theorem of Calculus. However, we need to use limit if an endpoint of the interval is special (infinite).

$$$\int_{0}^{\infty}\left( x e^{- x^{2}} \right)dx=\lim_{x \to \infty}\left(- \frac{e^{- x^{2}}}{2}\right)-\left(- \frac{e^{- x^{2}}}{2}\right)|_{\left(x=0\right)}=\frac{1}{2}$$$

Do not forget about the minus sign: $$$\int_{\infty}^{0}\left( x e^{- x^{2}} \right)dx=-\int_{0}^{\infty}\left( x e^{- x^{2}} \right)dx=- \frac{1}{2}$$$

Answer: $$$\int_{\infty}^{0}\left( x e^{- x^{2}} \right)dx=- \frac{1}{2}=-0.5$$$

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Kalkylator för bestämda och oegentliga integraler

Beräkna bestämda och oegentliga integraler steg för steg

Solution