Answer
$ -\displaystyle \frac{1}{4x+10}+C$
Work Step by Step
see Substitution RuIe, p.962:
1. Write $u$ a{\it s} a function of x.
2. Take the derivative $du/dx$ and solve for the quantity $dx$ in terms of $du$.
3. Use the expression you obtain in step 2 to substitute for $dx$ in the given integral and substitute $u$ for its defining expression.
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(1) Given $\quad u=2x+5,$
(2)$ \displaystyle \quad du=2dx\ \ \Rightarrow\ \ dx=\frac{du}{2}$
$(3)$
$\displaystyle \int(2x+5)^{-2}dx=\int u^{-2}\cdot\frac{du}{2}$=
... constant multiple ...
$=\displaystyle \frac{1}{2}\int u^{-2}du$= ... power rule ...
$=\displaystyle \frac{1}{2}\cdot\frac{u^{-1}}{-1}+C$ ... bring x back ...
$=-\displaystyle \frac{(2x+5)^{-1}}{2}+C$
$= -\displaystyle \frac{1}{2(2x+5)}+C$
$=-\displaystyle \frac{1}{4x+10}+C$
