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Ch 7.2 Wkst AP Calculus BC Name: Integration by Parts Objective: to integrate when u-substitution does not work Ex 1: x 3e x dx Ex 2: Use the Tabular Method to evaluate x 3 sin x dx Ex 3: x 2 ln x dx Ch 7.2 Wkst AP Calculus BC Name: Integration by Parts For problems #1 to #7, find the integrals by Integration by Parts. 1. x 2. 2 x e dx x 3 cos 2 x dx 3. x 4. 4 ln x dx ln x x10 dx 5. arctan x dx 6. e x cos x dx 7. x arcsin x 2 dx Hint: Do a normal u-substitution first, then integrate by parts. 8. sec 3 x dx Hint: Write as sec x sec ² x first, then integrate by parts. You’ll also need to do a trigonometric substitution using the identity 1 + sec ² x = tan ² x. 9. Given the initial condition y(0) 4 , find the particular solution to the differential equation dy x sin x . dx 2 y 10. dy 1 ) , then when x = 2 the solution (3x 2 ln x) y 2 and its solution curve passes through ( 1 , dx 2 curve passes through y = If (A) 1 4 ln 2 (B) 1 4 2 ln 2 (C) 1 4 ln 2 (D) 1 4 2 ln 2 (E) 1 6 2 ln 2 11. dy x e x and f (0) 2 . Find all values of x in the interval [ 0 , 3 ] for which the particular dx solution y f (x) equals its average value. CALCULATOR REQUIRED. Suppose (A) 1.501 (B) 1.624 (C) 1.998 (D) 2.615 (E) 2.332 ANSWERS: 1) x2 e x 2x e x 2 e x C 2) 1 x 3 sin 2 x 3 x 2 cos 2 x 3 x sin 2 x 3 cos 2 x C 2 4 4 8 1 x 5 ln x 1 x 5 C 5 25 3) 4) 1 x 9 ln x 1 x 9 C 9 81 5) x arctan x 1 ln 1 x 2 C 2 6) 1 e x sin x 1 e x cos x C 2 2 7) 1 x 2 arcsin x 2 1 1 x 4 C 2 2 1 sec x tan x 1 ln sec x tan x C 2 2 8) 9) 10) D 11) C y x cos x sin x 16