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  • Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.18 Question 10 Maths Textbook Solution.

Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.18 Question 10 Maths Textbook Solution.

Answers (1)

Answer: \log \left|\sin ^{2} x+2+\sqrt{\sin ^{4} x+4 \sin ^{2} x-2}\right|+c

Hint Let \sin ^{2} x=t

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

Explanation:

              \int \frac{\sin 2 x}{\sqrt{\sin ^{4} x+4 \sin ^{2} x-2}} d x.................(1)

            Let  \sin ^{2} x=t

            2 \sin x \cos 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}+4 t-2}}

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

              =\int \frac{d t}{\sqrt{(t+2)^{2}-6}}

              Let t+2 = u                                                                        (Differentiate w.r.t to u)

              \mathrm{dt}=\mathrm{du}

               =\int \frac{d u}{\sqrt{u^{2}-6}}

               =\log \left|u+\sqrt{u^{2}-6}\right|+c

               =\log \left|t+2+\sqrt{(t+2)^{2}-6}\right|+c

               =\log \left|\sin ^{2} x+2+\sqrt{\sin ^{4} x+4 \sin ^{2} x-2}\right|+c\left[\because \int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\log \left|x+\sqrt{x^{2}-a^{2}}\right|+c\right]

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