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Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 23 Maths Textbook Solution.

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


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


To solve this equation we use sin x formula and substitute method.


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

                                                                        \left[\sin 2 x=(\cos x+\sin x)^{2}-1, \quad t=\cos x+\sin x, \quad d t=(-\sin x+\cos x) d x\right.

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

Let  t=\cos x+\sin x, \quad d t=(-\sin x+\cos x) d x

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

=-\log \left|t+\sqrt{t^{2}-1}\right|+c

=-\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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