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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 $$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( x \sin{\left(x^{2} \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{x \sin{\left(x^{2} \right)} d x}=- \frac{\cos{\left(x^{2} \right)}}{2}$$$ (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.
$$$\left(- \frac{\cos{\left(x^{2} \right)}}{2}\right)|_{\left(x=\frac{\pi}{2}\right)}=- \frac{\cos{\left(\frac{\pi^{2}}{4} \right)}}{2}$$$
$$$\left(- \frac{\cos{\left(x^{2} \right)}}{2}\right)|_{\left(x=- \frac{\pi}{2}\right)}=- \frac{\cos{\left(\frac{\pi^{2}}{4} \right)}}{2}$$$
$$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( x \sin{\left(x^{2} \right)} \right)dx=\left(- \frac{\cos{\left(x^{2} \right)}}{2}\right)|_{\left(x=\frac{\pi}{2}\right)}-\left(- \frac{\cos{\left(x^{2} \right)}}{2}\right)|_{\left(x=- \frac{\pi}{2}\right)}=0$$$
Answer: $$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( x \sin{\left(x^{2} \right)} \right)dx=0$$$