`int sqrt(1 - 4x^2) dx` Evaluate the integral

`intsqrt(1-4x^2)dx`

`=intsqrt(1-(2x)^2)dx`

Now apply the integral substitution,

Let u=2x,

`=>du=dx`

`=intsqrt(1-u^2)du`

Now using the standard integral,

`intsqrt(a^2-x^2)dx=(xsqrt(a^2-x^2))/2+a^2/2arcsin(x/a)+C`

`=(usqrt(1-u^2))/2+1/2arcsin(u/1)+C`

Now substitute back u=2x,

`=(2xsqrt(1-(2x)^2))/2+1/2arcsin((2x)/1)+C`

`=xsqrt(1-4x^2)+1/2arcsin(2x)+C`