Question: If I<sub>1</sub> = $\int_{0}^{\pi/2}{\frac{x}{\sin x}dx}$ and I<sub>2</sub> = \(\int_{0}^{1}{\frac...

Question

If I₁ = $\int_{0}^{\pi/2}{\frac{x}{\sin x}dx}$ and I₂ = $\int_{0}^{1}{\frac{\tan^{- 1}x}{x}dx}$ , then $\frac{I_{1}}{I_{2}}$ =

Answer 1

I₂ = $\int_{0}^{1}\frac{\tan^{- 1}x}{x}dx$ , x = tanq

Ž I₂ = $\int_{0}^{\pi/4}\frac{2\theta}{\sin 2\theta}d\theta$ = $\frac{1}{2}\int_{0}^{\pi/2}\frac{x}{\sin x}dx$ = $\frac{1}{2}.I_{1}$

Ž $\frac{I_{1}}{I_{2}}$ = 2

Question: If I<sub>1</sub> = \(\int_{0}^{\pi/2}{\frac{x}{\sin x}dx}\) and I<sub>2</sub> = \(\int_{0}^{1}{\frac...