Let u = x dx =cos x, dv = cosxdx du dx =1 v = xdx ∫cos v = sin x the formula dv du We have ∫x cos xdx = x sin x ∫sin x + dx – = x sin x + cos x + c Example 3 Find x3 ln x dx ∫ Solution x3 ln xdx = ln x. x3 dx writing in the order LIATE ∫ ∫ Let u = ln x dv = x3 1 du  x dx x 4 3 4 Substituting in the formula dv du v dx we have dx 36 1 x x 3 4 4 x4 3 ln x 4 dx x3 4 x4  ln  x c 4 16 Example 4
Let u = x dx =cos x, dv = cosxdx du dx =1 v = ∫cos xdx v = sin x the formula dv du We have ∫x cos xdx = x sin x – ∫sin x + dx = x sin x + cos x + c Example 3 Find ∫ x3 ln x dx Solution ∫ x3 ln xdx = ∫ ln x. x3 dx writing in the order LIATE Let u = ln x dv = x3 du dx  1 x 3 x 4 4 Substituting in the formula dv v du dx dx we have 36 3 x 4 x 4 1 3 x4 4 ln x x3 4 dx  x4 4 ln x 16  c Example 4
CALCULUS%20II.DOCX