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

December 2019 1 213 Report

2) Calcule a seguinte integral definida:

$\int\limits^2_1 { \frac{v^3+3v^5}{v^4} } \, dv$

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gabrieldoile Temos o seguinte:

$\int\limits^2_1 { \frac{v^3 + 3v^5}{v^4} } \, dv$

Assim:

$\int { \frac{v^3 + 3v^5}{v^4} } \, dv = \int { \frac{v^3}{v^4} + \frac{3v^5}{v^4} } \, dv = \int { \frac{1}{v} + 3v } \, dv = \int { \frac{1}{v} } \, dv + \int { 3v } \, dv \\ \\ 
 \int { \frac{1}{v} } \, dv + \int { 3v } \, dv = ln(v) + 3 \cdot \frac{v^2}{2} + C$

Aplicando os limites:

$(ln(v) + 3 \cdot \frac{v^2}{2} )\limits^2_1 \\ \\ 
= (ln(2) + 3 \cdot \frac{2^2}{2}) - (ln(1) + 3 \cdot \frac{1^2}{2} ) \\ \\ 
= (ln(2) + 6) - (0 + \frac{3}{2} ) \\ \\ 
= ln(2) + 6 - \frac{3}{2} \\ \\ 
= ln(2) + \frac{9}{2}$

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