Question

qual a integral?
$\int\limits$x sen8x dx

Answer 1

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Integral por partes

$\huge\boxed{\boxed{\boxed{\boxed{\displaystyle\sf\int\! u\!\cdot\! dv=u\cdot v-\int\! v\!\cdot\! du}}}}$

$\underline{\rm para~resolver~esta~quest\tilde ao~vamos~usar~o~macete~do~ILATE}\\\large\boxed{\begin{array}{c}\sf I\longrightarrow inversa~trigonom\acute etrica\\\sf L\longrightarrow\ell ogaritmo\\\sf A\longrightarrow aritm\acute etica\\\sf T\longrightarrow trigonom\acute etrica\\\sf E\longrightarrow exponencial\\\tt vale~lembrar~que~isto~se~trata\\\tt de~um~crit\acue \acute erio~para~escolha~do~"u"\\\tt visando~facilitar~o~c\acute alculo~da~integral\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm fac_{\!\!,}a}\\\sf u=x\longrightarrow du=dx\\\sf dv=sen(8x)dx\longrightarrow v=-\dfrac{1}{8}cos(8x)\\\displaystyle\sf\int u\!\cdot\!dv=u\cdot v-\int\! v\cdot\!du\\\displaystyle\sf\int x\cdot\! sen(8x)\!~dx= x\cdot\bigg(-\dfrac{1}{8}cos(8x)\bigg)-\int-\dfrac{1}{8}cos(8x)\cdot\!~dx\\\displaystyle\sf\int x\!\cdot sen(8x)\!~dx=-\dfrac{1}{8}x\cdot cos(8x)+\dfrac{1}{8}\int cos(8x)\!~dx\end{array}}$

$\boxed{\begin{array}{l}\displaystyle\sf\int\! x\cdot\! sen(8x)\!~dx=-\dfrac{1}{8}x\cdot cos(8x)+\dfrac{1}{8}\cdot\dfrac{1}{8}sen(8x)+k\\\boxed{\boxed{\boxed{\boxed{\displaystyle\sf\int\! x\cdot sen(8x)\!~dx=\dfrac{1}{8}\cdot\bigg[-x\cdot cos(8x)+\dfrac{1}{8}sen(8x)\bigg]+k}}}}\end{array}}$

Answer 2

Resposta:

∫xsen8x dx = 1/64sen8x - 1/8cos8x + c

Explicação passo-a-passo:

Segue resolução por imagem.