Question

Calcule a integral definida abaixo:
$\int\limits \frac{y+5 y^{3} }{ y^{4} }$

Answer 1

$\displaystyle\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+C~~\forall n\neq-1~~\left(pois~\dfrac{d}{dx}\left[\dfrac{x^{n+1}}{n+1}+C\right]=x^{n}\right)\\\\\\\int \dfrac{1}{x}dx=ln|x|+C~~~\left(pois~\dfrac{d}{dx}[ln|x|+C]=\dfrac{1}{x}\right)$
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$\displaystyle\int\dfrac{y+5y^{3}}{y^{4}}dy$

Dividindo o numerador e o denominador por y:

$\displaystyle\int\dfrac{y+5y^{3}}{y^{4}}dy=\int\dfrac{1+5y^{2}}{y^{3}}dy\\\\\\\int\dfrac{y+5y^{3}}{y^{4}}dy=\int\left(\dfrac{1}{y^{3}}+5\dfrac{y^{2}}{y^{3}}\right)dy\\\\\\\int\dfrac{y+5y^{3}}{y^{4}}dy=\int\dfrac{1}{y^{3}}dy+\int5\dfrac{1}{y^{1}}dy\\\\\\\int\dfrac{y+5y^{3}}{y^{4}}dy=\int y^{-3}dy+5\int\dfrac{1}{y}dy\\\\\\\int\dfrac{y+5y^{3}}{y^{4}}dy=\dfrac{y^{-3+1}}{-3+1}+5\cdot ln|y|+C\\\\\\\boxed{\boxed{\int\dfrac{y+5y^{3}}{y^{4}}dy=-\dfrac{y^{-2}}{2}+5\cdot ln|y|+C}}$