Question

Integrate:sinxcos(x-1/4.pi)dx

Answer 1

I guess maybe you mean
sin x cos(x-pi/4)dx

sin x cos y = .5( sin x +sin y) + .5 (sin x - sin y)
(remember "beat frequency" )
so

.5(sin x + sin(x-pi/4)dx
+
.5(sin x - sin(x-pi/4)dx
----------------------------
add (oh my)
sin x dx

-cos x + c

Answer 2

I guess maybe you mean
sin x cos(x-pi/4)dx

sin x cos y = .5 sin (x+y) + .5 sin (x-y)
(remember "beat frequency" )
so

.5sin ( 2x-pi/4)dx
+
.5sin (pi/4)dx

Answer 3

note sin pi/4 = sqrt2/2

-.25 cos (2x-pi/4) - .25 sqrt 2

Answer 4

note we no longer have sine curves with zero mean value but now have an average plus a cosine curve with twice the frequency.

Answer 5

-.25 cos (2x-pi/4) - .25 x sqrt 2 + c