3) Calcule a seguinte integral definida:.... Pergunta de ideia deLucasJairo

December 2019 1 156 Report

3) Calcule a seguinte integral definida:

$\int\limits^1_0 {(3+x \sqrt{x} )} \, dx$

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gabrieldoile Temos o seguinte:

$\int\limits^1_0 { 3 + x \sqrt{x} } \, dx$

Assim:

$\int { 3 + x \sqrt{x} } \, dx = \int { 3 + \sqrt{x^3} } \, dx = \int { 3 + x^{ \frac{3}{2} } } \, dx = \int { 3 } \, dx + \int {x^{ \frac{3}{2} } } \, dx \\ \\ 
 \int { 3 } \, dx + \int {x^{ \frac{3}{2} } } \, dx = 3x + \frac{x^{ \frac{3}{2} + 1 }}{ \frac{3}{2} + 1 } = 3x + \frac{x^{ \frac{5}{2} }}{ \frac{5}{2} } = 3x + \frac{2}{5} \cdot \sqrt{x^5} + C$

Logo aplicando os limites:

$(3x + \frac{2}{5} \cdot \sqrt{x^5} )\limits^1_0 \\ \\ 
=(3 \cdot 1 + \frac{2}{5}\cdot \sqrt{1^5} ) - (3\cdot 0 + \frac{2}{5} \cdot \sqrt{0^5} ) \\ \\ 
= (3 + \frac{2}{5} ) - 0 \\ \\ 
= \frac{17}{5}$

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