Question

If `int_0^(2a)1/(1+4x^2)dx=π/6`, then the value of a is ______.

Options

• `sqrt3/4`

• `sqrt3/2`

• `sqrt3`

• `2sqrt3`

Answer 1

If `int_0^(2a)1/(1+4x^2)dx=π/6`, then the value of a is `bbunderline(sqrt3/4)`.

Explanation:

`int_0^(2a)1/(1+4x^2)dx=π/6`

`1/(1+4x^2)=1/(1+(2x)^2)`

∴ k = 2

`int_0^(2a)1/(1+(2x)^2)dx=[1/2tan^-1(2x)]_0^(2a)`

= `1/2tan^-1(2*2a)-1/2tan^-1(0)`

= `1/2tan^-1(4a)-0`

= `1/2tan^-1(4a)`

`1/2tan^-1(4a)=pi/6`

tan^-1(4a) = `pi/3`

4a = tan `(pi/3)`

4a = `sqrt3`

a = `sqrt3/4`