Example6: Find: òx² sinx dx

We use : òu . dv = u . v – òv . du

let u = x² Then du = 2x . dx , let dv = sin x . dx Then v = òsin x . dx = – cos x

  òu . dv = u . v – òv . du

 òx² . sin x . dx = x² . (– cos x)  – ò( – cos x ) . 2x dx

 òx² . sin x . dx = – x² cos x) + ò2x cos x dx  ...................... (1)

To integrate: ò2x cos x dx

let u = 2x Then du = 2 . dx , let dv = cos x . dx Then v = òcos x . dx = sin x

  òu . dv = u . v – òv . du

 ò2x . cos x . dx = 2x .sin x – òsin x . 2 dx

 ò2x . cos x . dx = 2x .sin x – ò2sin x dx

 ò2x . cos x . dx = 2x .sin x – 2( – cos x )

 ò2x . cos x . dx = 2x .sin x + 2 cos x  ...........................(2)

From (2) In (1) Then

 òx² . sin x . dx = – x² cos x + 2x .sin x + 2 cos x + c