Question

Evaluate the following integrals using properties of integration:

`int_0^pi(xsinx)/(1 + sinx)  "'d"x`

Answer 1

Let f(x)= `sinx/(1 + sinx)`

`"f"(pi - x) = (sin(pi - x))/(1 + sin(pi - x))`

= `sinx/(1 + sinx)`

= f(x)

`int_0^"a"  xf(x) "d"x = "a"/2 int_0^"a" f(x)  "d"x`

If `f("a" - x) = f(x)`

`int_0^pi xsinx/(1 + sinx)  "d"x = pi/2 int_0^pi  sinx/(1 + sin x)  "d"x`

= `pi/2 int_0^pi  (sin x(1 - sin x))/((1 + sin x)(1 - sin x)) "d"x`

= `pi/2int_0^pi  (sinx  sin^2x)/(1 - sin^2x)  "d"x`

= `pi/2 int (sin x - sin^2x)/(cos^x)  "d"x`

= `pi/2[int_0^pi  sinx/(cos^2x)  "d"x  int_0^pi  (sin^2x)/(cos^2x)  "d"x]`

= `pi/2 [int_0^pi tan x sec x  dx - int_0^pi tan^2 x  "d"x]`

= `pi/2 [int_0^pi  tan x sec x  "d"x - int_0^pi (sec^2x - 1) "d"x]`

= `pi/2[[sec x]_0^pi - [tan x - x]_0^pi]`

= `pi/2 [(- 1 - 1) - (0 - pi - 0)]`

= `pi/2 [-2 + pi]`

= `pi/2 [pi- 2]`