定積分・広義積分計算機

この計算機は、定積分（すなわち上下限のある積分）を、広義積分も含めて、解法手順を示しながら計算を試みます。

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 $$$\int_{\frac{\pi}{4}}^{\pi}\left( \sin{\left(x \right)} \right)dx$$$

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

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

$$$\left(- \cos{\left(x \right)}\right)|_{\left(x=\frac{\pi}{4}\right)}=- \frac{\sqrt{2}}{2}$$$

$$$\int_{\frac{\pi}{4}}^{\pi}\left( \sin{\left(x \right)} \right)dx=\left(- \cos{\left(x \right)}\right)|_{\left(x=\pi\right)}-\left(- \cos{\left(x \right)}\right)|_{\left(x=\frac{\pi}{4}\right)}=\frac{\sqrt{2}}{2} + 1$$$

Answer: $$$\int_{\frac{\pi}{4}}^{\pi}\left( \sin{\left(x \right)} \right)dx=\frac{\sqrt{2}}{2} + 1\approx 1.70710678118655$$$

Please try a new game Rotatly

定積分・広義積分計算機

定積分と広義積分を手順を追って計算する

Solution