∫ x+2 /√( x ^ 2 +2x+3 ) #160; dx #160; #160; #160;let #160; x^2 +2x +3 =t #160; #160; =∫ x+1+1 /√( x ^ 2 +2x+3 ) #160; dx ...

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Integration by Substitution

∫x+2√x2+2x+3d...

Question

$\int \frac{x + 2}{\sqrt{x^{2} + 2 x + 3}} d x$

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Solution

$\int \frac{x + 2}{\sqrt{x^{2} + 2 x + 3}} d x$ let $x^{2} + 2 x + 3 = t$
$= \int \frac{x + 1 + 1}{\sqrt{x^{2} + 2 x + 3}} d x$ $⟹ (2 x + 2) d x = 2 t d t$ $⟹ (x + 1) d x = t d t$
$= \int \frac{x + 1}{\sqrt{x^{2} + 2 x + 3}} d x$ $+ \int \frac{1}{\sqrt{x^{2} + 2 x + 3}} d x$
$= \int \frac{t}{t} d t$ $+ \int \frac{1}{\sqrt{{(x + 1)}^{2} + 2}} d x$
$= t + ln ∣ ∣ (x + 1) + \sqrt{(2 + (x + 1)^{2})} ∣ ∣ + C$
$= (x^{2} + 2 x + 3) + ln ∣ ∣ (x + 1) + \sqrt{(2 + (x + 1)^{2})} ∣ ∣ + C$

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