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정적분 및 가적분 계산기
정적분과 광의적분을 단계별로 계산하세요
이 계산기는 단계별 풀이를 보여 주면서 상한과 하한이 있는 정적분(진정적분 포함)을 계산하려고 시도합니다.
Solution
Your input: calculate $$$\int_{0}^{\pi}\left( \sin{\left(x^{2} \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\sin{\left(x^{2} \right)} d x}=\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}$$$ (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.
$$$\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=\pi\right)}=\frac{\sqrt{2} \sqrt{\pi} S\left(\sqrt{2} \sqrt{\pi}\right)}{2}$$$
$$$\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=0\right)}=0$$$
$$$\int_{0}^{\pi}\left( \sin{\left(x^{2} \right)} \right)dx=\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=\pi\right)}-\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=0\right)}=\frac{\sqrt{2} \sqrt{\pi} S\left(\sqrt{2} \sqrt{\pi}\right)}{2}$$$
Answer: $$$\int_{0}^{\pi}\left( \sin{\left(x^{2} \right)} \right)dx=\frac{\sqrt{2} \sqrt{\pi} S\left(\sqrt{2} \sqrt{\pi}\right)}{2}\approx 0.772651712690066$$$