integral `(e^(ax) cos(bx+c))`

Explanation

`"Let I"=int e^(ax).cos(bx+c)dx`
`=e^(ax)int cos(bx+c)dx-int[(d)/(dx)(e^(ax))int cos(bx+c)dx]dx`
`=e^(ax). (sin(bx+c))/(b)-fae^(ax).(sin(bx+c))/(b)dx`
`=(1)/(b)e^(ax)/sin(bx+c)-(a)/(b)int e^(ax).sin(bx+c)dx`
`=(1)/(b)e^(ax).sin(bx+c)-(a)/(b)[e^(ax)intsin(bx+c)dx-int{(d)/(dx)(e^(ax))int sin(bx+c)dx}dx]`
`=(1)/(b)e^(ax).sin(bx+c) (a)/(b)[e^(ax).({-cos(bx+c)})/(b)-int ae^(ax).({-cos (bx+c)})/(b)dx]`
`=(1)/(b)e^(ax).sin(bx+c)+(a)/(b^(2))e^(ax).cos(bx+c)-(a^(2))/(b^(2))int e^(ax).cos (bx+c)dx`
`therefore I=(1)/(b)e^(ax).sin(bx+c)+(a)/(b^(2))e^(ax).cos(bx+c)-(a^(2))/(b^(2)).I`
`therefore (1+(a^(2))/(b^(2)))I=(1)/(b)e^(ax).sin(bx+c)+(a)/(b^(2))e^(ax).cos(bx+c)`
`therefore ((b^(2)+a^(2))/(b^(2)))I=(e^(ax))/(b^(2))[b sin (bx+c)+a cos (bx+c)]`
`therefore I=(e^(ax))/(a^(2)+b^(2))[a cos (bx+c)+b sin (bx+c)]+c_(1)`