|
|
|
|
|
|
|
|
|
|
|
|
Calcolatore di integrali definiti e impropri
Calcola integrali definiti e impropri passo dopo passo
Il calcolatore cercherà di valutare l'integrale definito (cioè con estremi), inclusi quelli impropri, mostrando i passaggi.
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: $$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( x \sin{\left(x^{2} \right)} \right)dx=0$$$