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INTEGRATION BY SUBSTITUTION

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Appreciate when an algebraic substitution is required to determine an integral, Integrate functions that require an algebraic substitution. Determine definite integrals where an algebraic substitution is required

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  • April 12, 2022
  • 6
  • 2020/2021
  • Exam (elaborations)
  • Questions & answers
  • general power rule
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kwazithwala
KG THWALA MATE1B1 2020
Assignment HW5 due 09/09/2020 at 11:59pm SAST



This last integral is: = +C
1. (1 point) Find (Leave out constant of integration from your answer.)
Z
After substituting back for u we obtain the following final
F(x) = x(x2 + 4)6 dx
form of the answer:
Give a specific function for F(x). = +C
(Leave out constant of integration from your answer.)
Answer(s) submitted:
F(x) = •
Answer(s) submitted: •
• [[xˆ(2) +4]ˆ(7)]/[14] •

(correct) •
Correct Answers:
• (x*x+4)**(6+1)/(2*(6+1))
(incorrect)
Correct Answers:
2. (1 point) • cos(6*t)
Evaluate the indefinite integral. • -1/(6*sin(6*t))
• -u**11/6
(arcsin x)4
Z
√ • -u**(11+1)/72
dx
1 − x2 • -(cos(6*t))**(12)/72

Answer: +C 5. (1 point) Note: You can get full credit for this problem by
Answer(s) submitted: just answering the last question correctly. The initial questions
• [arcsinˆ(5)(x)]/5 are meant as hints towards the final answer and also allow you
(correct) the opportunity to get partial credit.Z
12
Correct Answers: Consider the indefinite integral x7 7 + 12x8 dx
• (arcsin(x))ˆ
Then the most appropriate substitution to simplify this integral
3. (1 point) is
Evaluate the indefinite integral. u=
Then dx = f (x) du where
e2x
Z
f (x) =
dx = +C
e4x + 64 Z After making the substitution we obtain the integral
Answer(s) submitted:
g(u) du where
• [arctan[(eˆ(2x))/(8)]]/16
(correct) g(u) =
Correct Answers: This last integral is: = +C
• (1/(2*8))*arctan(eˆ(2 * x)/8) (Leave out constant of integration from your answer.)
After substituting back for u we obtain the following final
4. (1 point) Note: You can get full credit for this problem by form of the answer:
just entering the answer to the last question correctly. The initial = +C
questions are meant as hints towards the final answer and also (Leave out constant of integration from your answer.)
allow you the opportunity to get partial
Z credit. Answer(s) submitted:
Consider the indefinite integral cos11 (6t) sin(6t) dt •
Then the most appropriate substitution to simplify this integral •

is

u= Then dt = f (t) du where •
f (t) =
(incorrect)
Z After making the substitution we obtain the integral Correct Answers:
g(u) du where • 7+12*xˆ(7+1)
g(u) = • 1/(12*(7+1)*xˆ7)
1

, • uˆ12/(12*(7+1)) •
• uˆ(12+1)/(12*(7+1)*(12+1)) •
• (7+12*xˆ(7+1))ˆ(12+1)/(12*(7+1)*(12+1)) •

6. (1 point) Note: You can get full credit for this problem by •
just answering the last question correctly. The initial questions •
are meant as hints towards the final answer and also allow you
(incorrect)
the opportunity to get partial credit.R
1√ Correct Answers:
Consider the indefinite integral 5x+7 x
dx
• sin(z)
Then the most appropriate substitution to simplify this inte- • 1/cos(z)
gral is • 1/uˆ8
u= • 0.5
Then dx = f (x) du where • 1
f (x) = • 18.1428571428571
AfterRmaking the substitution and simplifying we obtain the
8. (1 point)
integral g(u) du where
Evaluate the definite integral.
g(u) =
Z e4
This last integral is: = +C dx
(Leave out constant of integration from your answer.) 1 x(1 + ln x)
After substituting back for u we obtain the following final
form of the answer: Answer(s) submitted:
= +C •
(Leave out constant of integration from your answer.) (incorrect)
Answer(s) submitted: Correct Answers:
• • 1.6094379124341

• 9. (1 point) Find the following indefinite integrals.
x
Z
• √ dx = +C
• x +2
cos(t)
Z
(incorrect) dt = +C
Correct Answers: (2 sin(t) + 5)2
Answer(s) submitted:
• 5*sqrt(x)+7
• 2*sqrt(x)/5 •
• 2/(5*u) •
• 2*ln(u)/5 (incorrect)
• 2*ln(5*sqrt(x)+7)/5 Correct Answers:
7. (1 point) Note: You can get full credit for this problem by • 2*(x + 2)*sqrt(x + 2)/3 - 2*2*sqrt(x + 2)
• -1/(2*(2*sin(t) + 5))
just answering the last question correctly. The initial questions
are meant as hints towards the final answer and also allow you 10. (1 point) Note: You can get full credit for this problem
the opportunity to get partial credit. by just entering the answer to the last question correctly. The
cos(z)
Z π/2
Consider the definite integral dz initial questions are meant as hints towards the final answer and
8 also allow you the opportunity to get partial credit.
π/6 sin (z)
Then the most appropriate substitution to simplify this inte- Consider the indefinite integral
gral is Z
8
u= dx
8 + ex
Then dz = f (z) du where
f (z) = The most appropriate substitution to simplify this integral is
After making the substitution and simplifying we obtain the u = f (x) where
Z b f (x) =
integral g(u) du where We then have
a
g(u) = dx = g(u) du
a= where
b= g(u) =
This definite integral has value = Hint: you need to back substitute for x in terms of u for this
Answer(s) submitted: part.
2

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