³³ 2 - korpisworld maximus...worksheet 5.4—integration by parts show all work. no calculator...

Calculus Maximus WS 5.4: Integration by Parts Page 1 of 4 Name_________________________________________ Date________________________ Period______ Worksheet 5.4—Integration by Parts Show all work. No calculator unless stated. Multiple Choice 1. If 2 cos 2 sin x xdx hx x xdx ³ ³ , then hx (A) 2sin 2 cos x x x C (B) 2 sin x x C (C) 2 2 cos sin x x x x C (D) 4cos 2 sin x x x C (E) 2 2 cos 4sin x x x C 2. sin 5 x x dx ³ = (A) cos 5 sin 5 x x x C (B) 1 cos 5 sin 5 5 25 x x x C (C) 1 cos 5 sin 5 5 5 x x x C (D) 1 cos 5 sin 5 5 25 x x x C (E) 5 cos 5 sin 5 x x x C 3. 2 csc x xdx ³ (A) 3 csc 6 x x C (B) cot ln sin x x x C (C) cot ln sin x x x C (D) cot ln sin x x x C (E) 2 sec tan x x x C

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Page 1: ³³ 2 - korpisworld Maximus...Worksheet 5.4—Integration by Parts Show all work. No calculator unless stated. Multiple Choice 1. If ³³xxdxhx xxdxcos 2 sin 2 , then hx (A) 2sin

Calculus Maximus WS 5.4: Integration by Parts

Page 1 of 4

Name_________________________________________ Date________________________ Period______ Worksheet 5.4—Integration by Parts Show all work. No calculator unless stated. Multiple Choice 1. If � �2 cos 2 sinx xdx h x x xdx �³ ³ , then � �h x

(A) 2sin 2 cosx x x C� � (B) 2 sinx x C� (C) 22 cos sinx x x x C� � (D) 4cos 2 sinx x x C� � (E) � �22 cos 4sinx x x C� � �

2. � �sin 5x x dx³ =

(A) � � � �cos 5 sin 5x x x C� � � (B) � � � �1cos 5 sin 55 25x x x C� � � (C) � � � �1cos 5 sin 5

5 5x x x C� � �

(D) � � � �1cos 5 sin 55 25x x x C� � (E) � � � �5 cos 5 sin 5x x x C� �

3. 2cscx xdx ³

(A) 3csc

6x x C� (B) cot ln sinx x x C� � (C) cot ln sinx x x C� � �

(D) cot ln sinx x x C� � � (E) 2sec tanx x x C� � �

Page 2: ³³ 2 - korpisworld Maximus...Worksheet 5.4—Integration by Parts Show all work. No calculator unless stated. Multiple Choice 1. If ³³xxdxhx xxdxcos 2 sin 2 , then hx (A) 2sin

Calculus Maximus WS 5.4: Integration by Parts

Page 2 of 4

4. The graph of � �y f x conforms to the slope field for

the differential equation 4 lndy x xdx

, as shown. Which

of the following could be � �f x ?

(A) 2 22 ln 3x x � (B) 3 ln 3x x � (C) 2 22 ln 3x x x� � (D) � �22 3 ln 1x x� �

(E) 3

2 4ln2 ln 33xx x � �

Short Answer 5. Evaluate the following integrals.

(a) xxe dx�³ (b) � �2 sinx x dxS³ (c) 1sin xdx�³ (d) 2ln xdx³ (e) arctan 4tdt³

Page 3: ³³ 2 - korpisworld Maximus...Worksheet 5.4—Integration by Parts Show all work. No calculator unless stated. Multiple Choice 1. If ³³xxdxhx xxdxcos 2 sin 2 , then hx (A) 2sin

Calculus Maximus WS 5.4: Integration by Parts

Page 3 of 4

6. Evaluate the following definite integrals. Show the antiderivative. Verify on your calculator.

(a) 0

sin 3t tdtS

³ (b) � �1

1 xx e dx��³

lne x dxx³

(d) 1 3

20 4

r drr�

³ (Hint: let 3 2r r r � )

7. Solve: 2secdy x xdx

and 1y when 0x .

8. Find the area of the region enclosed by the x-axis and the curve siny x x for 2xS Sd d .

Page 4: ³³ 2 - korpisworld Maximus...Worksheet 5.4—Integration by Parts Show all work. No calculator unless stated. Multiple Choice 1. If ³³xxdxhx xxdxcos 2 sin 2 , then hx (A) 2sin

Calculus Maximus WS 5.4: Integration by Parts

Page 4 of 4

9. A slowing force, symbolized by the “Dashpot” in the figure at right, slows

the motion of the weighted spring so that the mass’s position at time t is given by 2 costy e t� , 0t t . Find the average position of the mass on the interval > @0,2t S� . Give an exact answer, then verify on your calculator.

10. Using u-substitution and then integration by parts, evaluate sin xdx³ .