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Exam (elaborations)

Exam (elaborations) WTW 158: Calculus (WTW158)

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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.

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  • August 17, 2022
  • 5
  • 2022/2023
  • Exam (elaborations)
  • Questions & answers

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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

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