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
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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