Question

Por favor me ajudem calculo 2 integrais indefinida. Cos(x)√2+sen(x) dx

Answer 1

Calcular a integral indefinida:

   $\mathsf{\displaystyle\int cos(x)\sqrt{2+sen(x)}\,dx}\\\\\\ \mathsf{\displaystyle=\int \sqrt{2+sen(x)}\cdot cos(x)\,dx}$

Faça uma substituição de variáveis:

   $\mathsf{2+sen(x)=u\quad\Longrightarrow\quad cos(x)\,dx=du}$

e a integral fica

   $\mathsf{\displaystyle=\int \sqrt{u}\,du}\\\\\\ \mathsf{\displaystyle=\int u^{1/2}\,du}\\\\\\ \mathsf{=\dfrac{u^{(1/2)+1)}}{\frac{1}{2}+1}+C}\\\\\\ \mathsf{=\dfrac{u^{3/2}}{\frac{3}{2}}+C}\\\\\\ \mathsf{=\dfrac{2}{3}\,u^{3/2}+C}\\\\\\ \mathsf{=\dfrac{2}{3}\,\sqrt{u^3}+C}\\\\\\ \mathsf{=\dfrac{2}{3}\sqrt{\big(2+sen(x)\big)^3}+C\quad\longleftarrow\quad resposta.}$

Dúvidas? Comente.

Bons estudos! :-)