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Integrate the function
$\int \frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}}=I(\operatorname{say}) x \in[0,1]$
$\int \frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}}=I(\operatorname{say}) x \in[0,1]$
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$\mathrm{I}=\int \frac{\sin ^{-1} \sqrt{\mathrm{x}}-\frac{\pi}{2}+\sin ^{-1} \sqrt{\mathrm{x}}}{\pi / 2} \mathrm{dx}$
$=\frac{2}{\pi} \int 2 \sin ^{-1} \sqrt{\mathrm{x}} \mathrm{dx}-\int 1 \mathrm{dx} \quad \ldots(i)$
Let $\mathrm{I}_1=\sin ^{-1} \sqrt{\mathrm{x}} \mathrm{dx}$ Put $\sqrt{\mathrm{x}}=\sin \theta$
$\Rightarrow \mathrm{dx}=2 \sin \theta \cos \theta \mathrm{d} \theta=\sin 2 \theta \mathrm{d} \theta$
$\therefore \mathrm{I}_1=\int \theta \sin 2 \theta \mathrm{d} \theta$
$=\theta \int \sin 2 \theta-\int 1\left(\frac{-\cos 2 \theta}{2}\right) d \theta$
$=\frac{-\theta \cos 2 \theta}{2}+\frac{1}{2} \frac{\sin 2 \theta}{2}$
$=\frac{-\sin ^{-1} \sqrt{\mathrm{x}}(1-2 \mathrm{x})}{2}+\frac{1}{4} \times(2 \sqrt{\mathrm{x}} \sqrt{1-\mathrm{x}})+\mathrm{C}$
$=\frac{2}{\pi} \int 2 \sin ^{-1} \sqrt{\mathrm{x}} \mathrm{dx}-\int 1 \mathrm{dx} \quad \ldots(i)$
Let $\mathrm{I}_1=\sin ^{-1} \sqrt{\mathrm{x}} \mathrm{dx}$ Put $\sqrt{\mathrm{x}}=\sin \theta$
$\Rightarrow \mathrm{dx}=2 \sin \theta \cos \theta \mathrm{d} \theta=\sin 2 \theta \mathrm{d} \theta$
$\therefore \mathrm{I}_1=\int \theta \sin 2 \theta \mathrm{d} \theta$
$=\theta \int \sin 2 \theta-\int 1\left(\frac{-\cos 2 \theta}{2}\right) d \theta$
$=\frac{-\theta \cos 2 \theta}{2}+\frac{1}{2} \frac{\sin 2 \theta}{2}$
$=\frac{-\sin ^{-1} \sqrt{\mathrm{x}}(1-2 \mathrm{x})}{2}+\frac{1}{4} \times(2 \sqrt{\mathrm{x}} \sqrt{1-\mathrm{x}})+\mathrm{C}$
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