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प्रश्न
Evaluate the following integrals using properties of integration:
`int_0^pi(xsinx)/(1 + sinx) "'d"x`
उत्तर
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]`
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