Question

Integrate the following with respect to x :

`cosx/(cos(x - "a"))`

Answer 1

`int cosx/(cos(x - "a"))  "d"x`

Put `x - "a"` = u

x = `"u" + "a"`

dx = du

`int cosx/(cos(x - "a"))  "d"x  = int (cos("u" + "a"))/(cos "u")  "du"`

= `int (cos "u" cos "a" - sin "u" sin "a")/(cos "u")  "du"`

= `int ((cos "u" cos "a")/(cos "u") - (sin"u" sin"a")/cos"u") "du"`

= `int cos "a" " du" - int tan "u" sin "a" * "du"`

= `cos "a" int  "du" - sin "a" int  tan "u"  "du"`

= `cos "a"("u") - sin "a" log |sec "u"| + "c"`

`int cosx/(cos(x - "a"))  "d"x  = (x - "a") cos"a" - sin"a" log|sec(x - "a")| + "c"`