find the indefinite integral. ∫(2x + 5)^(-2)dx ∫(2x + 5)^(-2)dx = □

Explanation:

Step1: Use substitution method

Let $u = 2x+5$, then $du=2dx$, and $dx=\frac{1}{2}du$.

Step2: Rewrite the integral

The integral $\int(2x + 5)^{-2}dx$ becomes $\frac{1}{2}\int u^{-2}du$.

Step3: Integrate $u^{-2}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), for $n=-2$, we have $\int u^{-2}du=\frac{u^{-2 + 1}}{-2+1}+C=-\frac{1}{u}+C$.

Step4: Substitute back $u = 2x+5$

$\frac{1}{2}\int u^{-2}du=-\frac{1}{2u}+C=-\frac{1}{2(2x + 5)}+C$.

Answer:

$-\frac{1}{2(2x + 5)}+C$