Exam (elaborations)
MAT2691 EXAM PACK 2023
- Course
- Institution
LATEST QUESTIONS WITH ANSWERS
[Show more]
Seller
Follow
reinah44
Reviews received
Content preview
MAT2691EXAM
PACK 2023
LATEST QUESTIONS
AND ANSWERS
,QUESTION 1.1
Determine the following integrals
∫ ( 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ) √2𝑎 3 + 3𝑏𝑥 2 + 6𝑐𝑥 + 𝑑 𝑑𝑥
SOLUTION
Let 𝑢 = 2𝑎𝑥 + 3𝑏𝑥 + 6𝑐𝑥 + 𝑑
Hint:
3 2
𝑑𝑢
= 6𝑎𝑥 2 + 6𝑏𝑥 + 6𝑐
𝑑𝑥 The main concept to maste
in Que 1.1 is the basic
integration. Since 𝑎𝑥 2 +
𝑑𝑢 = (𝑎𝑥
6 2 + 6𝑏𝑥 + 𝑐) 𝑏𝑥 + 𝑐 is a derivative of
𝑑𝑥
2𝑎 3 + 3𝑏𝑥 2 + 6𝑐𝑥 + 𝑑
𝑑𝑢
Therefore 𝑑𝑥 = makes it easy to integrate
6(𝑎𝑥 2 +𝑏𝑥+𝑐
using substitution.
𝑑𝑢
∫ ( 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ) √𝑢
6(𝑎𝑥 2 + 𝑏𝑥 + 𝑐)
1
∫ √ 𝑢 𝑑𝑢
6
3
1 2
= × 𝑢2 + 𝑐
6 3
2 3
= 𝑢 +𝑐
18 2
1 3 2 3
= √(2𝑎𝑥 + 3𝑏𝑥 + 6𝑐𝑥 + 𝑑) +𝑐 ANS.
9
1.2
√𝑡𝑎𝑛 2 2𝑥 − 9
∫ 𝑑𝑥
𝑐𝑜𝑠 2 2𝑥
1
=∫ √𝑡𝑎𝑛 2 2𝑥 − 9 𝑑𝑥
𝑐𝑜𝑠 2 2𝑥
= ∫ 𝑠𝑒𝑐 2 √𝑡𝑎𝑛 2 2𝑥 − 9 𝑑𝑥
= ∫ 𝑠𝑒𝑐 2 𝑥 √(tan 2𝑥) 2 −3 2 𝑑𝑥
From the standard
2 (𝑥) (𝑥 )
= ∫ ′(𝑥) √ [𝑓(𝑥)] 2 − 𝑎 2 𝑑𝑥 = − 𝑎 𝑎𝑟𝑐 cos ℎ ( 𝑓 + 𝑓 √[𝑓(𝑥)] 2 − 𝑎 2 + C
2 𝑎 2
Since the derivative of 𝑡𝑎𝑛2𝑥 = 𝑠𝑒𝑐 2 2𝑥
, 2|Page MAT2691 OCT/NOV2010
𝑡𝑎𝑛 2 2𝑥−9 𝑑𝑥
∫√ 𝑐𝑜𝑠 2 2𝑥
=− 9 𝑎𝑟𝑐 cos ℎ 𝑡𝑎𝑛2𝑥 + 𝑡𝑎𝑛2𝑥 √ (𝑡𝑎𝑛2𝑥 ) − 9) + 𝐶
4 3 4
1.3
∫
𝑠𝑖𝑛ℎ𝑥
𝑑𝑥
Hint:
1 + 𝑐𝑜𝑠ℎ𝑥
Que 1.3. The derivative of 1+
From
𝑐𝑜𝑠ℎ𝑥 is 𝑠𝑖𝑛ℎ𝑥
𝑓′(𝑥) Since now we know that the
∫ = 𝑙𝑛 | 𝑓(𝑥) | + 𝑐 question consist of the function and
𝑓(𝑥)
its derivative, therefore one of the
Since the derivative of 1 + cosh 𝑥 = sinh 𝑥 standard formulae has to be used
i.e.
=∫ 𝑠𝑖𝑛ℎ𝑥 𝑑𝑥 𝑙𝑛= |1 + cos ℎ 𝑥 | + 𝑐 ANS.
1+𝑐𝑜𝑠ℎ𝑥
1.4
𝜋
4 Que 1.4 To be able to integrate this
∫ 𝑠𝑖𝑛 3 𝑥. 𝑐𝑜𝑠 3 𝑥 𝑑𝑥 question easily the function has to
0
𝜋
be simplified first. Usually with this
type of questions we have to
∫04 𝑠𝑖𝑛 3 𝑥( ( 𝑐𝑜𝑠 2 𝑥 )( 𝑐𝑜𝑠𝑥 ) )𝑑𝑥 From { 𝑐𝑜𝑠 2 𝑥 = 1 − 𝑠𝑖𝑛 2 𝑥} simplify the function and leave it in
𝜋
the form
= ∫04 𝑠𝑖𝑛 3 𝑥((1 − 𝑠𝑖𝑛 2 x)(cosx))dx
𝜋
= ∫04 𝑠𝑖𝑛 3 𝑥𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛 5 𝑥 𝑐𝑜𝑠 𝑥 𝑑𝑥 In this form it can easily be solved
𝜋 𝜋 using the standard equations.
𝑠𝑖𝑛 4 𝑥 𝑠𝑖𝑛 6 𝑥
4 4
=[ ]0 −[ ]
4 6 0
𝜋
4 𝜋 𝑠𝑖𝑛 6 ( )
4
= [𝑠𝑖𝑛 ( )− ]
4 6
1
=
24
=0.042 ANS.
1.5
𝑑𝑥
∫
𝑥2 + 4𝑥 + 20
, By completing the square on the denominator.
3|Page MAT2691 OCT/NOV2010
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller reinah44. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.00. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75323 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now