find the intergral

Question

if f is continuous and ∫0 to 5 f(x)dx=12, then ∫0 to pi/2 f(5sin(t))(cos(t))dt=?

Answer 1

x = 5*sin t ; then dx = 5* cos T dT --> (1/5) dx = cos T dT

f(5 * sin t) * cos(t) dt = (1/5) * f(x) * dx

when x=0, 0 = 5 * sin t ---> 0 = sin t ---> t = 0

when x=5, 5 = 5 * sin t --> 1 = sin t ---> t = pi/2

Therefore, integral f(5 * sint)* cost dt , t = 0...pi/2 =

1/5 * integral f(x) dx, t = 0...5

= 12/5