Use the Substitution Formula,
Integral from a to b f left parenthesis g left parenthesis x right parenthesis right parenthesis times g prime left parenthesis x right parenthesis dx equals Integral from g left parenthesis a right parenthesis to g left parenthesis b right parenthesis f left parenthesis u right parenthesis du∫abf(g(x))•g′(x) dx=∫g(a)g(b)f(u) du
where
g(x)equals=u,
to evaluate the following integral.
Integral from 0 to StartFraction pi Over 2 EndFraction StartFraction 2 Bold sin left parenthesis 2 t right parenthesis Over 4 minus Bold cos left parenthesis 2 t right parenthesis EndFraction dt∫0π22 sin (2t)4−cos (2t)dt.