Question

If `int_0^π f(sinx)dx = kint_0^π f(sinx)dx`, then find the value of k.

Answer 1

I = `int_0^π xf(sinx)dx`  ....(1)

∴ I = `int_0^π f(π - x) f[sin(π - x)]dx`

∴ I = `int_0^π (π - x) f(sin x)dx`  ....(2)

Adding (1) and (2)

2I = `int_0^π f(sin x)[x + π - x]dx`

∴ I = `int_0^π π/2 f(sin x)dx`

= `k int_0^π f(sin x)dx`  ....(Given)

∴ k = `π/2`