Question

Evaluate the indefinite integral.

integral of 4e^(4x)sin[e^(4x)]dx

Answer 1

let y = 4 x
dy = 4 dx
so
integral of e^y sin y dy

try e^y cos y
derivative = -e^y sin y + e^y cos y
so
try ae^y cos y + be^y sin y
derivative = -ae^ysiny+ae^ycosy+be^ycosy+be^ysiny
we need a=-b to get rid of cos terms
so
-b e^y cos y +b e^y sin y
so
be^y sin y - be^y cos y
try derivative
b e^y cos y+be^y sin y+be^y siny-be^y cosy
=2 b e^y sin y
so let b = 1/2 and we are there


=(1/2)e^y [ sin y-cos y] + c