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Step-by-step Solution

Integral of (2x+3)/(x^2+6x+9)

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Answer

$2\ln\left|x+3\right|+\frac{3}{x+3}+C_0$

Step-by-step explanation

Problem to solve:

$\int\:\frac{2x+3}{\:x^2+6x+9}dx$
1

The trinomial $\frac{2x+3}{x^2+6x+9}$ is perfect square, because it's discriminant is equal to zero

$\Delta=b^2-4ac=6^2-4\left(1\right)\left(9\right) = 0$
2

Using the perfect square trinomial formula

$a^2+2ab+b^2=(a+b)^2,\:where\:a=\sqrt{x^2}\:and\:b=\sqrt{9}$

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Answer

$2\ln\left|x+3\right|+\frac{3}{x+3}+C_0$

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$\int\:\frac{2x+3}{\:x^2+6x+9}dx$

Main topic:

Integration by substitution

Used formulas:

9. See formulas

Time to solve it:

~ 0.7 seconds