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Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.18 Question 1 Maths Textbook Solution.

Answers (1)

Answer:  \frac{1}{2} \log \left|x^{2}+\sqrt{x^{4}+a^{4}}\right|+c

Hint:  \int \sqrt{x^{2}+a^{2}}=\log \left|x+\sqrt{x^{2}+a^{2}}\right|+c

Given: \int \frac{x}{\sqrt{x^{4}+a^{4}}} d x               .............(1)



                    Let x^{2}=t

                    2 x d x=d t

                    x d x=\frac{d t}{2}                                                    (Differentiate w.r.t to t)

        Put in (1)

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

                       =\frac{1}{2} \log \left|t+\sqrt{t^{2}+a^{4}}\right|+c                

                        =\frac{1}{2} \log \left|x^{2}+\sqrt{x^{4}+a^{4}}\right|+c


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