Question

Resolva a Integral Indefinida:

a) integral de (1 - cos (t/2))^2 * sen (t/2) dt

Answer 1

Resolver a integral indefinida:

$\displaystyle\int\!\left[1-\cos\!\left(\frac{t}{2}\right)\right]^2\mathrm{sen}\!\left(\frac{t}{2}\right)\!dt\\\\\\ =\int\!2\cdot \dfrac{1}{2}\left[1-\cos\!\left(\frac{t}{2}\right)\right]^2\mathrm{sen}\!\left(\frac{t}{2}\right)\!dt\\\\\\ =2\int\!\left[1-\cos\!\left(\frac{t}{2}\right)\right]^2\cdot \dfrac{1}{2}\,\mathrm{sen}\!\left(\frac{t}{2}\right)\!dt\qquad\mathbf{(i)}$


Faça a seguinte substituição:

$1-\cos\!\left(\dfrac{t}{2}\right)=u~\Rightarrow~\dfrac{1}{2}\,\mathrm{sen}\!\left(\dfrac{t}{2}\right)\!dt=du$


Substituindo em $\mathbf{(i),}$ a integral fica

$=\displaystyle 2\int\!u^2\,du\\\\\\ =2\cdot \dfrac{u^{2+1}}{2+1}+C\\\\\\ =2\cdot \dfrac{u^3}{3}+C\\\\\\ =\dfrac{2}{3}\,u^3+C\\\\\\ =\dfrac{2}{3}\cdot\left[1-\cos\!\left(\dfrac{t}{2}\right)\right]^3+C\qquad\checkmark$


Caso tenha problemas para visualizar a resposta, experimente abrir pelo navegador: http://brainly.com.br/tarefa/7483531


Dúvidas? Comente.


Bons estudos! :-)


Tags: integral indefinida substituição mudança variável trigonométrica cosseno cos seno sen cálculo integral