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Step-by-step Solution

Integral of (x+3)/(x^2+4x+5)

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Answer

$\frac{1}{2}\ln\left|\left(x+2\right)^2+1\right|-2arctan\left(x+2\right)+3arctan\left(x+2\right)+C_0$

Step-by-step explanation

Problem to solve:

$\int\frac{x+3}{x^2+4x+5}dx$
1

Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$

$\int\frac{x+3}{x^2+4x+5+4-4}dx$
2

Factor the perfect square trinomial $x^2+4x+4$

$\int\frac{x+3}{\left(x+2\right)^2+1}dx$

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Answer

$\frac{1}{2}\ln\left|\left(x+2\right)^2+1\right|-2arctan\left(x+2\right)+3arctan\left(x+2\right)+C_0$
$\int\frac{x+3}{x^2+4x+5}dx$

Main topic:

Integration by substitution

Used formulas:

9. See formulas

Time to solve it:

~ 1.25 seconds

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