Definite integral by substitution

Question

Why isn't the answer (d) but (b)? Thank you for your help!!

Answer 1

You get a factor of two from the square root upon subbing $x=(u-1)/2$, but also a factor of $1/2$ from $dx=du/2$. These cancel each other out. Also, the limits in $u$ are $2(-1/2)+1=0$ and $2(-1/4)+1=1/2$.

Answer 2

Set $u=cosx$ then $1-u^2=1-cos^2x=sin^2x$

Answer 3

You need to change the limits of the integration. Putting $x=-\frac{1}{2}$ and $x=-\frac{1}{4}$ into $u=2x+1$ will give you the limits in (b).