If `int_0^a1/(1+4x^2)dx=\ pi/8` , then `a` equals `pi/2` b. `pi/4` c. `1/2` d. `1`

Explanation

`frac{1}{2} `

`int_{0}^{alpha} frac{1}{1+4 x^{2}} d x=frac{pi}{8} `

`Rightarrow int_{0}^{alpha} frac{1}{1+(2 x)^{2}} d x=frac{pi}{8} `

`Rightarrow frac{1}{2}[tan ^{-1} 2 x]_{0}^{alpha}=frac{pi}{8} `

`Rightarrow frac{1}{2} tan ^{-1} 2 alpha=frac{pi}{8} `

`Rightarrow 2 alpha=tan frac{pi}{4} `

`Rightarrow 2 alpha=1 `

`therefore alpha=frac{1}{2}`