Question

A integral indefinida ∫ (2x +1)^-1/2 dx com limite superior de integração 4 e limite inferior de integração 0 é

Answer 1

• O resultado dessa integral definida é igual a:

$\Large\displaystyle\text{ \begin{gathered}\sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=2\end{gathered} }$

Desejamos calcular a seguinte integral definida:

  $\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx\end{gathered} }$

Vamos então aplicar o método da substituição simples, chamando então:

$\Large\displaystyle\begin{gathered} \sf u=2x+1\end{gathered}$

$\Large\displaystyle\begin{gathered} \sf du=2 dx\end{gathered}$

$\Large\displaystyle\begin{gathered} \sf \frac{du}{2}= dx\end{gathered}$

Substituindo, temos que:

$\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=\int_0^4u^{-\frac{1}{2}}\frac{du}{2}\end{gathered} }$

E pela lineariedade:

$\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=\frac{1}{2}\cdot\int_0^4u^{-\frac{1}{2}}du\end{gathered} }$

Agora, vale ressaltar a seguinte propriedade de integração:

      $\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\sf\int x^ndx=\frac{x^{n+1}}{n+1} +\mathbb{C}\ , \ \forall n\neq-1}}\end{gathered} }$

Com isso, temos que:

$\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=\frac{1}{2}\cdot\left[\frac{u^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\right]\Bigg|_0^4\end{gathered} }$

$\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=\frac{1}{2}\cdot\left[2u^{\frac{1}{2}}\right]\bigg|_0^4\end{gathered} }$

$\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=\frac{1}{2}\cdot\left[2\sqrt{2x+1} \right]\bigg|_0^4\end{gathered} }$

Aplicando o Teorema Fundamental do Cálculo, temos que:

$\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=\frac{1}{2}\cdot\left[2\sqrt{2\cdot4+1} -2\sqrt{2\cdot0+1}\right]\end{gathered} }$

$\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=\frac{2\sqrt{9} -2\sqrt{1}}{2}\end{gathered} }$

$\Large\displaystyle\text{ \begin{gathered} \sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=\frac{4}{2}\end{gathered} }$

$\Large\displaystyle\text{ \begin{gathered} \therefore \green{\underline{\boxed{\sf\int_0^4\left(2x+1\right)^{-\frac{1}{2}}dx=2}}}\ \ \ (\checkmark).\end{gathered} }$

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