Question

integral sin (2x) dx from 0 to pi/8

Answer 1

\[\Large\bf\sf \int\limits_0^{\pi/8} \sin(\color{orangered}{2x})\;dx\]

Make the substitution:\[\Large\bf\sf \color{orangered}{u=2x}\]

Answer 2

Taking the derivative of our substitution:
\[\Large\bf\sf du=2\;dx\]Solving for dx, we'll divide each side by 2,\[\Large\bf\sf \frac{1}{2}du=dx\]

Answer 3

Then we just plug in our pieces.\[\Large\bf\sf \int\limits \sin(\color{orangered}{u})\;\frac{1}{2}du\]

Answer 4

Careful, our integral is now in terms of u, not x.
So we either have need to:
~Change the boundaries so they correspond to u
or
~After integrating, we can simply undo our substitution and use the original boundaries.