Answer:
[tex]\int\limits {e^{2s} cos\frac{s}{4} ds[/tex] =[tex]\frac{4 e^{2s} }{65 } ({8 cos (\frac{1}{4} ) s + sin \frac{1}{4} s} ))[/tex]
Step-by-step explanation:
Step(i):-
Given that [tex]f(s) = e^{2s} cos\frac{s}{4}[/tex]
Now integrating
[tex]\int\limits {f(s)} \, ds = \int\limits {e^{2s} cos\frac{s}{4} ds[/tex]
By using integration formula
[tex]\int\limits { e^{ax} cos b x dx = \frac{e^{ax} }{a^{2}+b^{2} } ( a cos b x + b sin b x )[/tex]
Step(ii):-
[tex]\int\limits {e^{2s} cos\frac{s}{4} ds[/tex] = [tex]\frac{e^{2s} }{(2)^{2}+(\frac{1}{4}) ^{2} } ( 2 cos (\frac{1}{4} ) s + \frac{1}{4} sin \frac{1}{4} s ))[/tex]
= [tex]\frac{e^{2s} }{(4+\frac{1}{16})} ( 2 cos (\frac{1}{4} ) s + \frac{1}{4} sin \frac{1}{4} s ))[/tex]
= [tex]\frac{e^{2s} }{(\frac{65}{16} } ( \frac{8 cos (\frac{1}{4} ) s + sin \frac{1}{4} s}{4} ))[/tex]
= [tex]16 X\frac{e^{2s} }{65 } ( \frac{8 cos (\frac{1}{4} ) s + sin \frac{1}{4} s}{4} ))[/tex]
=[tex]\frac{4 e^{2s} }{65 } ({8 cos (\frac{1}{4} ) s + sin \frac{1}{4} s} ))[/tex]
Final answer:-
[tex]\int\limits {e^{2s} cos\frac{s}{4} ds[/tex] =[tex]\frac{4 e^{2s} }{65 } ({8 cos (\frac{1}{4} ) s + sin \frac{1}{4} s} ))[/tex]
