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Write a Value of ∫ Sin X − Cos X √ 1 + Sin 2 X D X - Mathematics

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Sum

Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]

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Solution

\[\text{ Let I } = \int\frac{\left( \sin x + \cos x \right) dx}{\sqrt{1 - \sin 2x}}\]
\[ = \int\frac{\left( \sin x + \cos x \right) dx}{\sqrt{\sin^2 x + \cos^2 x - 2 \sin x \cos x}}\]
\[ = \int\frac{\left( \sin x + \cos x \right) dx}{\sqrt{\left( \sin x - \cos x \right)^2}}\]
\[ = \int\frac{\left( \sin x + \cos x \right) dx}{\left| \sin x - \cos x \right|}\]
\[ = \pm \int\left( \frac{\sin x + \cos x}{\sin x - \cos x} \right)dx\]
\[\text{ Let sin x} - \cos x = t\]
\[ \Rightarrow \left( \cos x + \sin x \right)dx = dt\]
\[ \therefore I = \pm \int\frac{dt}{t}\]
\[ = \pm \text{ ln }\left| t \right| + C\]
\[ = \pm \text{ ln} \left| \sin x - \cos x \right| + C \left( \because t = \sin x - \cos x \right)\]

Concept: Methods of Integration: Integration by Substitution
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Very Short Answers | Q 29 | Page 197
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