Before calculating the integral let us simplify the trigonometric expression using some trigonometric properties we have:
Applying the property of #cos# that says:
#cos(pi+alpha)=-cosalpha#
#cos(7x+pi)=cos(pi+7x)#
So,
#color(blue)(cos(7x+pi)=-cos7x)#
Applying two properties of #sin# that says:
#sin(-alpha)=-sinalpha#and
#sin(pi-alpha)=sinalpha#
We have:
#sin(5x-pi)=sin(-(pi-5x))=-sin(pi-5x)# since
#sin(-alpha)=-sinalpha#
#-sin(pi-5x)=-sin5x#
Since#sin(pi-alpha)=sinalpha#
Therefore,
#color(blue)(sin(5x-pi)=-sin5x)#
First Substitute the simplified answers then compute the integral:
#color(red)(intcos(7x+pi)-sin(5x-pi)#
#=int-cos(7x)-(-sin5x)#
#=int-cos7x+sin5x#
#=-intcos7x+intsin5x#
#color(red)(=-(sin7x)/7-(cos5x)/5+C# ( where #C #is a constant number).
