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Provide Solution For  R.D. Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.18 Question 17 Maths Textbook Solution.

Answers (1)

Answer: -\log |(\sin x+\cos x)+\sqrt{\sin 2 x}|+c

Hint  \sin ^{2} x+\cos ^{2} x=1

Given:\int \frac{\sin x-\cos x}{\sqrt{\sin 2 x}} d x


    \int \frac{\sin x-\cos x}{\sqrt{1+\sin 2 x-1}} d x

  \int \frac{\sin x-\cos x}{\sqrt{\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x-1}} d x                    \left[\begin{array}{l} \sin ^{2} x+\cos ^{2} x=1 \\ \sin 2 x=2 \sin x \cdot \cos x \end{array}\right]

\int \frac{\sin x-\cos x}{\sqrt{(\sin x+\cos x)^{2}-1}} d x                     ...........(1)            \left[(a+b)^{2}=a^{2}+b^{2}+2 a b\right]

Let \sin x+\cos x=t

(\cos x-\sin x) d x=d t

-(\sin x-\cos x) d x=d t

From (1) we have

         -\int \frac{d t}{\sqrt{t^{2}-1}}

       =-\log \left|t+\sqrt{t^{2}-1}\right|+c                                            \left[\because \int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\log \left|x+\sqrt{x^{2}-a^{2}}\right|+c\right]

       =-\log \left|\sin x+\cos x+\sqrt{(\sin x+\cos x)^{2}-1}\right|+c

        =-\log |\sin x+\cos x+\sqrt{\sin 2 x}|+c

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