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.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Solution

Your input: calculate $$$\int_{-1}^{1}\left( \operatorname{acos}{\left(x \right)} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\operatorname{acos}{\left(x \right)} d x}=x \operatorname{acos}{\left(x \right)} - \sqrt{1 - x^{2}}$$$ (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.

$$$\left(x \operatorname{acos}{\left(x \right)} - \sqrt{1 - x^{2}}\right)|_{\left(x=1\right)}=0$$$

$$$\left(x \operatorname{acos}{\left(x \right)} - \sqrt{1 - x^{2}}\right)|_{\left(x=-1\right)}=- \pi$$$

$$$\int_{-1}^{1}\left( \operatorname{acos}{\left(x \right)} \right)dx=\left(x \operatorname{acos}{\left(x \right)} - \sqrt{1 - x^{2}}\right)|_{\left(x=1\right)}-\left(x \operatorname{acos}{\left(x \right)} - \sqrt{1 - x^{2}}\right)|_{\left(x=-1\right)}=\pi$$$

Answer: $$$\int_{-1}^{1}\left( \operatorname{acos}{\left(x \right)} \right)dx=\pi\approx 3.14159265358979$$$

Määrättyjen ja epäoleellisten integraalien laskin

Laske määrättyjä ja epäoleellisia integraaleja askel askeleelta

Solution