Question

how to integrate
`e^(2s) *Cos (s)/(4)` , please help, I have no idea what to do, I would appreciate if you show your work with steps.

Answer 1

`\int\limits {e^(2s) cos(s)/(4) ds` =
`(4 e^(2s) )/(65 ) ({8 cos ((1)/(4) ) s + sin (1)/(4) s} ))`

Explanation:

Step(i):-

Given that
`f(s) = e^(2s) cos(s)/(4)`

Now integrating

`\int\limits {f(s)} \, ds = \int\limits {e^(2s) cos(s)/(4) ds`

By using integration formula

`\int\limits { e^(ax) cos b x dx = (e^(ax) )/(a^(2)+b^(2) ) ( a cos b x + b sin b x )`

Step(ii):-

`\int\limits {e^(2s) cos(s)/(4) ds` =
`(e^(2s) )/((2)^(2)+((1)/(4)) ^(2) ) ( 2 cos ((1)/(4) ) s + (1)/(4) sin (1)/(4) s ))`

=
`(e^(2s) )/((4+(1)/(16))) ( 2 cos ((1)/(4) ) s + (1)/(4) sin (1)/(4) s ))`

=
`(e^(2s) )/(((65)/(16) ) ( (8 cos ((1)/(4) ) s + sin (1)/(4) s)/(4) ))`

=
`16 X(e^(2s) )/(65 ) ( (8 cos ((1)/(4) ) s + sin (1)/(4) s)/(4) ))`

=
`(4 e^(2s) )/(65 ) ({8 cos ((1)/(4) ) s + sin (1)/(4) s} ))`

Final answer:-

`\int\limits {e^(2s) cos(s)/(4) ds` =
`(4 e^(2s) )/(65 ) ({8 cos ((1)/(4) ) s + sin (1)/(4) s} ))`