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.

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

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Solution

Your input: calculate $$$\int_{0}^{\pi}\left( \sin{\left(\phi \right)} \right)d\phi$$$

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

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

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

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

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

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Määrättyjen ja epäoleellisten integraalien laskin

Laske määrättyjä ja epäoleellisia integraaleja askel askeleelta

Solution