First slide

Methods of integration

Question

$\int \frac{dx}{\sqrt{x^{2} + 2 x + 2}}$ is equal to

Easy

A

$\log ∣ (x + 1) + \sqrt{x^{2} + 2 x + 2 ∣ + C}$
B

$\log ∣ x + \sqrt{(x^{2} + 2 x + 2) ∣ + C}$
C

$\log |x + 1 - \sqrt{x^{2} + 2 x + 2}| + C$
D

None of the above

Solution

Let
$\begin{array}{l} = \int \frac{dx}{\sqrt{x^{2} + 2 x + 2}} = \int \frac{dx}{\sqrt{(x^{2} + 2 x + 1) + 1}} \\ = \int \frac{1}{\sqrt{(x + 1)^{2} + (1)^{2}}} dx (let x + 1 = t) \end{array}$
$\Rightarrow dx = dt$
$\begin{array}{l} ∴ I = \int \frac{dt}{\sqrt{t^{2} + 1}} = \log |t + \sqrt{t^{2} + 1}| + C \\ [∵ \int \frac{dx}{\sqrt{x^{2} + a^{2}}} = \log |x + \sqrt{x^{2} + a^{2}}|] \\ = \log |(x + 1) + \sqrt{(x + 1)^{2} + 1}| + C \\ = \log |(x + 1) + \sqrt{x^{2} + 2 x + 2}| + C \end{array}$

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