Question

Evaluate the following integrals using properties of integration:

`int_0^(2pi) x log((3 + cosx)/(3 - cosx)) "d"x`

Answer 1

Let I = `int_0^(2pi) x log((3 + cosx)/(3 - cosx)) "d"x`

Let f(x) = `log((3 + cos x)/(3 - cos x))`

`"f"(2pi - x) = log((3 + cos(2pi - x))/(3 - cos(2pi - x)))`

= `log((3 + cosx)/(3 - cosx))`

= f(x)

∵ `int_0^"a" x f(x)  "d"x = "a"/2 int_0^"a" f(x)  "d"x  "if"  "f"("a" - x)` = f(x)

I = `(2pi)/2 int_0^(2pi) log((3 + cos x)/(3 - cos x)) "d"x`

= `2pi  int_0^pi log ((3 + cos x)/(3 - cos x)) "d"x`   .......(1)

I = `2pi int_0^pi log ((3 + cos (pi - x))/(3 - cos(pi - x))  "d"x`

`int_0^"a" f(x)  "d"x = int_0^"a" f("a" - x)  "d"x`

= `2 pi int_0^pi log((3 - cosx)/(3 + cos x)) "d"x` ........(2)

Add (1) and (2)

2I = `2pi  int_0^pi (log((3 + cosx)/(3 - cosx)) + log((3 - cosx)/(3 + cosx)))"d"x`

= `2pi  int_0^pi log((3 + cosx)/(3 - cosx) * (3 - cosx)/(3 + cosx)) "d"x`

2I = `2pi xx 0`  0

I = 0

`int_0^(2pi) x log ((3  cos x)/(3 - cos x))  "d"x` = 0