정적분 및 가적분 계산기

이 계산기는 단계별 풀이를 보여 주면서 상한과 하한이 있는 정적분(진정적분 포함)을 계산하려고 시도합니다.

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

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정적분 및 가적분 계산기

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Solution