How do you use substitution to integrate $(2x+3)/(x+7)^3$ ?

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Answer 1 · 2015-07-06T18:59:54

$int (2x+3)/((x+7)^3)\ dx=(-17-4x)/(2(x+7)^2)+C$

Let $u=x+7$ so that $du=dx$ and $x=u-7$ . Then $\int (2x+3)/((x+7)^3)\ dx=int (2(u-7)+3)/u^3\ du=int (2u^{-2}-11u^{-3})\ du$

$=-2u^{-1}+(11/2)u^{-2}+C$

$=(-2u+11/2)/u^2+C$

$=(-4(x+7)+11)/(2(x+7)^2)+C=(-17-4x)/(2(x+7)^2)+C$

How do you use substitution to integrate (2x+3)/(x+7)^3 (2x+3)/(x+7)^3 ?