Exam (elaborations)

Exam (elaborations) WTW 158: Calculus (WTW158)

54 views 2 purchases

Course
WTW 158: Calculus (WTW158)

Institution
University Of Pretoria (UP)

This document includes questions and step-by-step procedures on how to tackle differential equation questions. It will help you see a much easier method to use when dealing with such questions.

[Show more]

Preview 2 out of 5 pages

View example

Uploaded on August 17, 2022
Number of pages 5
Written in 2022/2023
Type Exam (elaborations)
Contains Questions & answers

differential equations
basic calculus
indefinite integrals
definite integrals
integration of trigonometric expressions
trigonometric substitution

Institution
University of Pretoria (UP)
Course
WTW 158: Calculus (WTW158)

CA$15.43

Added

Add to cart

Add to wishlist

100% satisfaction guarantee
Immediately available after payment
Both online and in PDF
No strings attached

BASIC CALCULUS
Quarter 4
Week 3&4
(I)Find the antiderivative of the following functions using substitution.
1. ∫ [ 2 e sin (x+ 1) cos ( x+1 ) ] dx
u=sin ⁡(x +1)

du=cos ( x +1 ) dx

∫ [ 2 e sin (x+ 1) cos ( x+1 ) ] dx=2 ∫ e u du
u
¿2e +c

substitute u=sin ( x +1 ) back intothe expression

∴∫ [ 2e cos ( x+1 ) dx ]=2e
sin ( x +1) sin ( x +1)
+c

2.∫ ( x −5) 2 xdx
2 5

2
u=x −5

du=2 xdx

∫ (x 2−5)5 2 xdx=∫ u5 du
1 6
¿ u +c
6
2
substitute u=x −5 back into the expression

1 2
∴∫ ( x −5 ) 2 xdx= ( x −5) +c
2 6
6

x2− x
3.∫ dx
1+3 x2 −2 x 3
2 3
u=1+ 3 x −2 x
2
du=6 x−6 x dx

¿−6 ( x −x ) dx
2

2
∫ 1+3xx−2 −2
x
x
3
−1 1
dx = ∫ du
6 u

, 1
¿− ln |u|+ c
6
2 3
substitute u=1+3 x −2 x

x2 −x −1
∴∫ dx= ln |1+3 x2 −2 x 3|+c
2
1+3 x −2 x 3
6
cosx
4.∫ 1+2 sinx dx

u=1+ 2 sinx

du=2 cosxdx

cosx 1 1
∫ 1+2 sinx dx= 2 ∫ u du
1
¿ ln |u|+ c
2
substitute u=1+2 sinx back intothe expression

cosx 1
∴∫ dx= ln |1+2 sinx|+c
1+2 sinx 2
5 3 2
3 x −2 x +5 x −2
5.∫ dx
x3 +1
5 3 2
3 x −2 x +5 x −2
Lets first simplify ∨break downthe expression 3
x +1

3 x 5−2 x 3 +5 x2−2 ( 3 x +5 x ) −(2 x + 2)
5 2 3

3
= 3
x +1 x +1
2 3 3
x (3 x +5) 2( x +1)
¿ −
3
x +1 (x¿¿ 3+ 1) ¿
2 3
x (3 x +5)
¿ 3
−2
x +1
5 3 2 2 3
3 x −2 x +5 x −2 x (3 x +5)
∫ 3
x +1
dx=∫ 3
x +1
dx −2∫ dx

|
2 3
x (3 x +5)
Lets ∫ 3
dx cause−2 ∫ dx=−2 x +c
x +1
3 3
u=x +1 ⟹ x =u−1
2
du=3 x dx

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 jacobsonny. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for CA$15.43. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

79202 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling