Question

10) Resolva a integral e diga o método que está usando:

$\int\limits {x~sin(x^2)} \, dx$

Answer 1

Resolvendo essa integral por substituição....

$\mathsf{\displaystyle\int~x~sin(x^{2})~dx}}}}$

Substituindo u = x², temos que du = 2x dx, veja como fica:

$\mathsf{\displaystyle\int~x~sin(x^{2})~dx}}}\\\\\\\\ \mathsf{\displaystyle\int~\dfrac{sin~u}{2}~du}}~~\textbf{Puxando~as~constantes~para~fora~da~integral,~teremos:}}}\\\\\\\\\ \mathsf{\dfrac{1}{2}~\displaystyle\int~sin~u~du}}}}}$

Usando a seguinte identidade:
$\mathsf{\displaystyle\int~sin{\theta}~d{\theta}}=\mathsf{-cos{\theta}}}}}}$

veja:

$\mathsf{\dfrac{1}{2}~\displaystyle\int~sin~u~du}}}\\\\\\\\\ \mathsf{\dfrac{1}{2}~(-cos~u)~du}}}}}$

Como u = x², temos que, por substituição definir:

$\mathsf{\dfrac{1}{2}~(-cos~u)~du}}}=\mathsf{-\dfrac{1}{2}~cos(x^{2})+C}}}}}\\\\\\\\\ \Large{\boxed{\boxed{\boxed{\boxed{\mathbf{~\therefore~\displaystyle\int~x~sin(x^{2})~dx}=\mathbf{-\dfrac{1}{2}~cos(x^{2})+C}}}}}}}}}}}}}}}}}}}~~\checkmark}}}$

Espero que te ajude. '-'

Answer 2

Integral por substituição simples.

$\displaystyle\mathsf{\int\,xsen({x}^{2})dx}=\mathsf{\dfrac{1}{2}\int~2xsen({x}^{2})dx}$

$\mathsf{u={x}^{2}\to~du=2x\,dx}$

Daí

$\displaystyle\mathsf{\int\,xsen({x}^{2})dx}=\mathsf{\dfrac{1}{2}\int~sen(u)du}\\=\mathsf{-\dfrac{1}{2}cos(u)+k}$

Portanto

$\displaystyle\mathsf{\int\,xsen({x}^{2})dx}=\mathsf{-\dfrac{1}{2}\,cos(2x)+k}$