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Integration of Irrational Algebraic Fractions - 1

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Question

Solve: $\int \frac{x + 1}{\sqrt{2 + 2 x + x^{2}}}$

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Solution

$\int \frac{x + 1}{\sqrt{2 + 2 x + x^{2}}} d x$
Let $2 + 2 x + x^{2} = t$
$(2 + 2 x) d x = d t$
$(x + 1) d x = \frac{d t}{2}$
$= \int \frac{d t}{2 \sqrt{t}}$
$= \sqrt{t} + c$
$= \sqrt{2 + 2 x + x^{2}} + c$

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