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STANDARD INTEGRATION
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Understand that integration is the reverse process of differentiation, Evaluate definite and indefinite integrals using basic integration rules.

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  • April 12, 2022
  • 4
  • 2020/2021
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KG THWALA MATE1B1 2020
Assignment HW6 due 09/18/2020 at 11:59pm SAST




1. (1 point) 4. (1 point)
Evaluate the indefinite integral. Evaluate the definite integral.
Z e8
dx
(arctan x)2
Z
dx 1 x(1 + ln x)
1 + x2
Answer(s) submitted:
Answer: +C • ln[81]/[2]
Answer(s) submitted: (correct)
• [[arctan(x)]ˆ(3)]/3 Correct Answers:
• 2.19722457733622
(correct)
Correct Answers: 5.
Z (1 point) Find the following indefinite integrals.
x
• (arctan(x))ˆ √ dx = +C
x +4
cos(t)
Z
2. (1 point) dt = +C
(4 sin(t) + 12)2
Evaluate the indefinite integral. Answer(s) submitted:
• [2(x-8)sqrt(x+4)]/3
e2x • -1/[16sin(t)+48]
Z
4x
dx
e +4 (correct)
Correct Answers:
• 2*(x + 4)*sqrt(x + 4)/3 - 2*4*sqrt(x + 4)
Answer = +C • -1/(4*(4*sin(t) + 12))
Answer(s) submitted: Z √
5
• [tanˆ(-1)[(eˆ(2x))/2]]/4 6. (1 point) ex dx = +C
Answer(s) submitted:
(correct)
Correct Answers: • 5eˆ(x/5)

• (1/(2*2))*arctan(eˆ(2 * x)/2) (correct)
Correct Answers:
• 5*eˆ(x/5)
3. (1 point) Find the indicated integrals.
ln(x5 ) 7. (1 point) Note: You can get full credit for this problem by
Z
(a) dx = +C
x just entering the answer to the last question correctly. The initial
Z t
e cos(e )t
questions are meant as hints towards the final answer and also
(b) dt = +C
4 + 4 sin(et )  allow you the opportunity to get partial credit.
Z 5/4 −1 4 Consider the indefinite integral
sin 5x
(c) √ dx =
25 − 16x2 9
Z
0
Answer(s) submitted: dx
9 + ex
• [5lnˆ(2)(x)]/2 The most appropriate substitution to simplify this integral is
• [ln(sin(eˆ(t))+1)]/4 u = f (x) where
• [piˆ(2)]/32
f (x) =
(correct) We then have
Correct Answers: dx = g(u) du
• 5*(ln(x))ˆ2/2 where
• ln(4+4*sin(eˆt))/4 g(u) =
• 0.308425137614586 Hint: you need to back substitute for x in terms of u for this
part.
1

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