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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.

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

Answers (1)

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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