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Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 9.

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Hint: We use indefinite integral formula then put limits to solve this integral.

Given: \int_{0}^{1}\frac{2x}{1+x^4}dx

Solution: I=\int_{0}^{1}\frac{2x}{1+x^4}dx

Put x^2=t

2x\; \; dx=dt

When x=0  then t=0  and

when x=1  then t=1

I=\int_{0}^{1}\frac{1}{t^2+1}dt                            \left[\int \frac{1}{1+x^{2}} d x=\tan ^{-1} \frac{x}{a}\right]

\begin{aligned} &=\left[\tan ^{-1}\left(\frac{t}{1}\right)\right]_{0}^{1} \mid \\ &=\left[\tan ^{-1} 1-\tan ^{-1} 0\right] \\ &=\frac{\pi}{4}-0=\frac{\pi}{4} \end{aligned}

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