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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.

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

Explanation:

    \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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