|
|
|
|
|
|
|
|
|
|
|
|
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 $$$\int_{f m n^{2} \nu s t y}^{\infty}\left( \sin{\left(x^{2} \right)} \right)dx$$$
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)
Since there is infinity in the upper bound, this is improper integral of type 1.
To evaluate an integral over an interval, we use the Fundamental Theorem of Calculus. However, we need to use limit if an endpoint of the interval is special (infinite).
$$$\int_{f m n^{2} \nu s t y}^{\infty}\left( \sin{\left(x^{2} \right)} \right)dx=\lim_{x \to \infty}\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)-\left(\frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)|_{\left(x=f m n^{2} \nu s t y\right)}=- \frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} f m n^{2} \nu s t y}{\sqrt{\pi}}\right)}{2} + \frac{\sqrt{2} \sqrt{\pi}}{4}$$$
Answer: $$$\int_{f m n^{2} \nu s t y}^{\infty}\left( \sin{\left(x^{2} \right)} \right)dx=- \frac{\sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} f m n^{2} \nu s t y}{\sqrt{\pi}}\right)}{2} + \frac{\sqrt{2} \sqrt{\pi}}{4}$$$