(i) Show that `int_(0)^(pi)xf(sinx)dx =(pi)/(2)int_(0)^(pi)f (sin x)dx`
(ii) Find the value of `int_(-1)^(3//2)|x sin pix|dx`.

Answer

(ii)` [(3pi+1)/(pi^(2))]`

Explanation

(i) Let `I= int _(0)^(pi) x f ( sin x) dx` . . . (i)
`rArr I= int _(0)^(pi) x (pi-x) f (sinx) dx` . . . (ii)
on adding Eqs . (i) and (ii) , we get
` 2 I = int _(0)^(pi) pi f ( sin x) dx`
`:. int _(0)^(pi) x f ( sin x) dx = (pi)/(2) int _(0)^(pi) f ( sin x) dx` Let `I = int _(-1)^(3//2) | x sin pix|dx`
Since , `|x sin pi x|={{:(x sin pi x_(,),-1 lt x le 1),(- x sin pi x_(,),1 lt x lt (3)/(2)):}`
`:. I = int _(-1)^(1) x sin pi x dx + int_(1) ^(3//2) - sin pi x dx`
`=2 [- ( x cos pix)/(pi) ]_(0)^(1)-2 int_(0)^(1) 1 * ((- cos pix)/(pi))dx`
`- {[ (-x cos pix)/(pi)]_(1)^(3//2)- int_(1)^(3//2)((- cos pix)/(pi))dx}`
`= 2 ((1)/(pi))+(2)/(pi) * [ (sinpix)/(pi) ]_(0)^(1) - ((1)/(pi)) - (1)/(pi) [ (sin pix)/(pi)]_(1)^(3//2)`
` =(2)/(pi) =(2)/(pi^(2))(0-0)+(1)/(pi) +(1)/(pi^(2))(+1-0)`
` = (3)/(pi) +(1)/(pi^(2))= ((3pi+1)/(pi^(2)))`