Take the integral:
integral cos(x) sinh(x) dx
For the integrand cos(x) sinh(x), integrate by parts, integral f dg = f g- integral g df, where
f = cos(x), dg = sinh(x) dx,
df = -sin(x) dx, g = cosh(x):
integral cos(x) sinh(x) dx = cos(x) cosh(x)+ integral sin(x) cosh(x) dx
For the integrand sin(x) cosh(x), integrate by parts, integral f dg = f g- integral g df, where
f = sin(x), dg = cosh(x) dx,
df = cos(x) dx, g = sinh(x):
integral cos(x) sinh(x) dx = sin(x) sinh(x)+cos(x) cosh(x)- integral cos(x) sinh(x) dx
Add integral cos(x) sinh(x) dx to both sides:
2 integral cos(x) sinh(x) dx = sin(x) sinh(x)+cos(x) cosh(x)+constant
Divide both sides by 2:
Answer: | integral cos(x) sinh(x) dx = 1/2 (sin(x) sinh(x)+cos(x) cosh(x))+constant
