Question

Integral
$\int\limits^ \sqrt{ \pi } } _ oxcos x^{2}dx$

Answer 1

$\int\limits_{0}^{\sqrt{\pi}}x*\cos(x^2)~dx$

$x^2=u$

$2x~dx=du$

$x~dx=\frac{1}{2}~du$

Agora para mudar os limites de integração

$x^2=u$

$(\sqrt{\pi})^2=u$

$u_{max}=\pi$

$u_{min}=0$

$\frac{1}{2}\int\limits_{0}^{\pi}\cos(u)~du$

$\left\frac{1}{2}\sin(u)\right]\limits_{0}^{\pi}$

$\frac{1}{2}[\sin(\pi)-sin(0)]$

$\boxed{\boxed{\there\int\limits_{0}^{\sqrt{\pi}}x*\cos(x^2)~dx=0}}$