Integral problems using U-substitution method. PDF

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We have a list pf useful problems for practice using u-substitution for students in calculus course. They would learn the method well....

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Integral

December 3, 2015

Problem 1 Evaluate the following indefinite integrals using the Substitution Rule. a.

R

c.

R

√ (x + 1) 2x + x2 dx

b.

R

eu du (1+eu )2

sin(2x) 1+cos2 (x) dx

d.

R

1+x dx 1+x2

Problem 2 Evaluate the following indefinite integrals by using the Substitution Rule. a.

R

c.

R

e.

R

5  x3 2 + x4 dx sec2



1/ x



x2

dx

eu du (1+eu )2

b.

R

(3t + 2)2.4 dt

d.

R

√ a+bx dx 3ax+bx3

f.

R

2

sin−1 (x) √ dx 1−x2

Problem 3 Evaluate the following definite integrals by using the Substitution Rule. a.

R1

c. e.

b.

R3

R2 √ x x − 1dx 1

d.

Ra √ 2 x a − x2 dx 0

R1

f.

R 1 ez +1

0

0

50

(3x − 1) dx

(1+

1 √ 4 dx x)

1

1 0 5x+1 dx

0 ez +z dz

Problem 4 √ R2 Evaluate −2 (x + 3) 4 − x2 dx by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.

Problem 5 Evaluate

R1 √ x 1 − x4 dx by making a substitution and interpreting the resulting integral in terms of an area. 0

Problem 6 If f is continuous on [0, π], use the substitution u = π − x to show. Z

π

xf (sin x) dx =

0

π 2

Z

π

f (sin x) dx 0

Problem 7 Sketch the region enclosed by each given curves and find its area. 1. y = x2 ,

y = 4x − x2

2. y = ex ,

y = xex ,

3. x = 2y2 ,

x = 4 + y2

4. y = |x| ,

y = x2 − 2

x=0

2...