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Provide Solution For R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.18 Question 11 Maths Textbook Solution.

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Answer: -\log \left|\left(\cos ^{2} x+\frac{1}{2}\right)+\sqrt{\cos ^{4} x+\cos ^{2} x+1}\right|+c

Hint Let \cos ^{2} x=t

Given: \int \frac{\sin 2 x}{\sqrt{\cos ^{4} x-\sin ^{2} x+2}} d x


            \int \frac{\sin 2 x}{\sqrt{\cos ^{4} x-\sin ^{2} x+2}} d x

            Let \cos ^{2} x=t

           -2 \cos x \sin x d x=d t

            -\sin 2 x d x=d t                                    (Differentiate w.r.t to t)

From (1) we have

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

  =-\int \frac{d t}{\sqrt{t^{2}+t+1}}

=-\int \frac{d t}{\sqrt{t^{2}+2 \cdot \frac{1}{2} \cdot t+\frac{1}{4}-\frac{1}{4}+1}}

                =\int \frac{d t}{\sqrt{\left(t+\frac{1}{2}\right)^{2}+\frac{3}{4}}}

                =-\log \left|t+\frac{1}{2}+\sqrt{\left(t+\frac{1}{2}\right)^{2}+\frac{3}{4}}\right|+c

                =-\log \left(\cos ^{2} x+\frac{1}{2}\right)+\sqrt{\cos ^{4} x+\cos ^{2} x+1} \mid+c

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