Question

Utilize o método de Substituição Simples para calcular a integral indefinida:

Answer 1

Resposta:

∫ √(2x+1)  dx  = ?

Substituindo u=2x+1   ==> du = 2dx

∫ √(u)  du/2

=(1/2) * u^(1/2+1) / (1/2+1)   + c

=(1/2) * u^(3/2) / (3/2)   + c

=(1/3) * u^(3/2) + c

Sabemos que u=2x+1

=(1/3) * (2x+1)^(3/2) + c

Letra E

Answer 2

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$\displaystyle\sf\int\sqrt{2x+1}~dx=\dfrac{1}{2}\int\sqrt{2x+1}\cdot2~dx\\\sf fac_{\!\!,}a~ u=2x+1\implies du=2~dx$

$\displaystyle\sf\int\sqrt{2x+1}~dx=\dfrac{1}{2}\int\sqrt{u}~du=\dfrac{1}{2}\int u^{\frac{1}{2}}~du=\dfrac{1}{\diagup\!\!\!\!2}\cdot\dfrac{\diagup\!\!\!\!2}{3}u^{\frac{3}{2}}+k\\\displaystyle\sf\int\sqrt{2x+1}=\dfrac{1}{3}(2x+1)^{\frac{3}{2}}+k$

$\large\boxed{\boxed{\boxed{\boxed{\displaystyle\sf\int\sqrt{2x+1}=\dfrac{(2x+1)^{3/2}}{3}+k}}}}$