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Need Solution for R.D. Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 51 Maths Textbook Solution.

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

Hint: Put  x^{2}=t

Given:\int x\left ( \tan ^{-1} x^{2}\right )dx

Solution:By letting x^{2}=t\Rightarrow 2xdx=dt\Rightarrow xdx=\frac{dt}{2}

               \therefore \int x\tan ^{-1}\left ( x^{2} \right )dx=\frac{1}{2}\int \tan ^{-1}t.1dt

                Integrate by parts, taking \tan ^{-1}t as the first function

                                         \begin{aligned} &=\frac{1}{2}\left[\tan ^{-1} t . t-\int \frac{1}{1+t^{2}} t d t\right] \\ &=\frac{1}{2}\left[t \tan ^{-1} t-\frac{1}{2} \int \frac{2 t}{1+t^{2}} d t\right] \\ &=\frac{1}{2}\left[t \tan ^{-1} t-\frac{1}{2} \log \left|1+t^{2}\right|\right]+c \\ &=\frac{1}{2} x^{2} \tan ^{-1} x^{2}-\frac{1}{4} \log \left(1+x^{4}\right)+c \end{aligned}

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