First slide

Methods of integration

Question

$Anti derivative of \frac{d x}{\sqrt{x^{2} + 2 x + 3}} is equal to$

Easy

A

$\log |(x + 1) + \sqrt{x^{2} + 2 x + 3}| + c$
B

$\log |(x + 1) + \sqrt{x^{2} + 2 x + 4}| + c$
C

$\log |(x + 1) - \sqrt{x^{2} + 2 x - 3}| + c$
D

$\log |(x + 1) + \sqrt{x^{2} - 2 x + 3}| + c$

Solution

$\int \frac{d x}{\sqrt{x^{2} + 2 x + 3}} = \int \frac{d x}{\sqrt{x^{2} + 2 x + 1 + 2}}$
$= \int \frac{d x}{\sqrt{(x + 1)^{2} + (\sqrt{2})^{2}}}$
$Use \int \frac{1}{\sqrt{x^{2} + a^{2}}} d x = \log |x + \sqrt{x^{2} + a^{2}}| + c$
$= \log |(x + 1) + \sqrt{x^{2} + 2 x + 3}| + c$

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