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Calculus II - 3 Problems on Homework | MATH 206, Assignments of Calculus

University of Hawaii (UH)Calculus

Material Type: Assignment; Class: Calculus II; University: University of Hawaii at Hilo; Term: Unknown 1989;

Typology: Assignments

2009/2010

Uploaded on 04/12/2010

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koofers-user-xdh 🇺🇸

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Math 206 - Calculus 2
Homework due September 2
Question 1. Compute the following definite integrals using u-substitutions.
(a) Zπ
0
3 cos2xsin x dx
(b) Z√3
−√3
4x
√x2+ 1
dx
(c) Z1
0
√x5+ 2x(5x4+ 2) dx
(d) Zπ/2
0
sin x
(3 + 2 cosx)2dx
(e) Zπ/4
0
(1 + etan θ) sec2θ dθ
Question 2. Use integration by parts to compute the following anti-derivatives.
(a) Zxsin x dx
(b) Zx3exdx
(c) Zxe5xdx
(d) Zx2e7xdx
(e) Z(x2
−5x)exdx
Question 3. In this question, we will compute
Zsin3x dx.
(a) Write sin3xas sin x·sin2xand then use the substitution sin2x= 1 −cos2x. What do you
get?
(b) In your answer from (a), distribute and separate your integrand into two separate integrals.
Solve these individual integrals and obtain an answer for Zsin3x dx.
(c) Check your answer by differentiating it and obtaining sin3x.
1

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Download Calculus II - 3 Problems on Homework | MATH 206 and more Assignments Calculus in PDF only on Docsity!

Homework due September 2Math 206 - Calculus 2

Question 1. Compute the following definite integrals using u-substitutions. (a)

∫ (^) π 0 3 cos

(^2) x sin x dx

(b)

−√ 3 √^4 x x^2 + 1 dx (c)

0

√x (^5) + 2x (5x (^4) + 2) dx

(d)

∫ (^) π/ 2 0

sin x (3 + 2 cos x)^2 dx (e)

∫ (^) π/ 4 0 (1 +^ e

tan θ) sec (^2) θ dθ

Question 2. Use integration by parts to compute the following anti-derivatives. (a)

x sin x dx (b)

x^3 ex^ dx (c)

xe^5 x^ dx (d)

x^2 e^7 x^ dx (e)

(x^2 − 5 x)ex^ dx

Question 3. In this question, we will compute ∫ sin^3 x dx. (a) Write sin^3 x as sin x · sin^2 x and then use the substitution sin^2 x = 1 − cos^2 x. What do you get? (b) In your answer from (a), distribute and separate your integrand into two separate integrals. Solve these individual integrals and obtain an answer for

sin^3 x dx. (c) Check your answer by differentiating it and obtaining sin^3 x. 1