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Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals1 Mark
Question
Evaluate: $\int \frac{d x}{\sqrt{1-2 x-x^2}}$
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Answer

$\text { (c) : Let } I=\int \frac{d x}{\sqrt{1-\left(x^2+2 x\right)}}=\int \frac{d x}{\sqrt{2-\left(x^2+2 x+1\right)}}$
$=\int \frac{d x}{\sqrt{2-(1+x)^2}}=\int \frac{d x}{\sqrt{(\sqrt{2})^2-(1+x)^2}}$
Put $1+x=z \Rightarrow d x=d z$
$\therefore I=\int \frac{d z}{\sqrt{(\sqrt{2})^2-z^2}}=\sin ^{-1} \frac{z}{\sqrt{2}}+C=\sin ^{-1}\left(\frac{1+x}{\sqrt{2}}\right)+C$

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