the result (2) in (1) 4 1 2 2x x e 2 2 e ) 1 1 1 2 2x 2x  2 2 4 1 2x  4 Example 5 Find x2 cos x dx ∫ Solution (In this problem Algebraic Function x2 comes before the Trigonometric function cos x. This satisfies the condition of dv Let u = x2 dx cos  x du dx v cos xdx sin x   2x  the formula we have dx dx  u dx uv v dx dv du cos xdx x sin x - sin x 2x dx  2 2  x sin x - 2 x sin xdx 2 (1) x 38 Now we must integrate x sin xdx Using again integration by parts. dv Let u = x dx sin  x du  1 sin xdx - cosx   v dx
the result (2) in (1) 2 2x 2 x e 1 2 4 e )  1 2 2x 1 2 2 1 2x 4  1 2x 4 Example 5 Find ∫ x2 cos x dx Solution (In this problem Algebraic Function x2 comes before the Trigonometric function cos x. This satisfies the condition of Let u = x2 dv dx  cos x du dx  2x v  cos xdx  sin x the formula u dv dx dx  uv v du dx dx we have x 2 cos xdx  x 2 sin x - sin x 2x dx  x 2 sin x - 2 x sin xdx (1) 38 Now we must integrate x sin xdx Using again integration by parts. Let u = x dv dx  sin x du dx  1 v  sin xdx  - cosx
CALCULUS%20II.DOCX