Määrättyjen ja epäoleellisten integraalien laskin

Laskin yrittää laskea määrätyn (eli rajoilla varustetun) integraalin, mukaan lukien epäoleelliset tapaukset, ja näyttää välivaiheet.

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.

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Solution

Your input: calculate $$$\int_{0}^{1}\left( x^{2} e^{x} \right)dx$$$

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

$$$\left(\left(x^{2} - 2 x + 2\right) e^{x}\right)|_{\left(x=1\right)}=e$$$

$$$\left(\left(x^{2} - 2 x + 2\right) e^{x}\right)|_{\left(x=0\right)}=2$$$

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

Answer: $$$\int_{0}^{1}\left( x^{2} e^{x} \right)dx=-2 + e\approx 0.718281828459045$$$

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Määrättyjen ja epäoleellisten integraalien laskin

Laske määrättyjä ja epäoleellisia integraaleja askel askeleelta

Solution