Question

Integral de 6x+7÷(x+2)^2 dx​

Answer 1

$\int{ \frac{6x + 7}{ {(x + 2)}^{2} } }dx \\ 6\int{ \frac{x}{ {(x + 2)}^{2} } }dx + 7\int{ \frac{dx}{ {(x + 2)}^{2} } }$

faça

u=x+2 →x=u-2

du=dx

$6\int{ \frac{u - 2}{ {u}^{2} } }du = 6\int{ \frac{1}{u}}du - 12\int{ \frac{du}{ {u}^{2} } }$

$6 ln|u| - 12.( - \frac{1}{u}) + c \\ 6 ln |u| + \frac{12}{u} + c \\ = 6 ln(x + 2) + \frac{12}{x + 2} + c$

$7\int{ \frac{dx}{ {(x + 2)}^{2} } } = 7\int{ \frac{du}{ {u}^{2} } } = - \frac{7}{u} + c \\ = - \frac{7}{x + 2} + c$

$\int{ \frac{6x + 7}{ {(x + 2)}^{2} } }dx \\ 6 ln(x + 2) + \frac{12}{x + 2} - \frac{7}{x + 2} + c \\ 6 ln(x + 2) + \frac{5}{x + 2} + c$