2.2 Exercises

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{{Mall:Ej vald flik|[[2.2 Variabelsubstitution|Teori]]}}
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{{Not selected tab|[[2.2 Substitution|Theory]]}}
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{{Mall:Vald flik|[[2.2 Övningar|Övningar]]}}
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{{Selected tab|[[2.2 Exercises|Exercises]]}}
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===Övning 2.2:1===
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===Exercise 2.2:1===
<div class="ovning">
<div class="ovning">
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Beräkna integralerna
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Calculate the integrals
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="100%"|<math>\displaystyle \int_{1}^{2} \displaystyle\frac{dx}{(3x-1)^4}\quad</math> genom att använda substitution <math>u=3x-1</math>
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|width="100%"|<math>\displaystyle \int_{1}^{2} \displaystyle\frac{dx}{(3x-1)^4}\quad</math> by using the substitution <math>u=3x-1</math>,
|-
|-
|b)
|b)
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|width="100%"| <math>\displaystyle \int (x^2+3)^5x \, dx\quad</math> genom att använda substitution <math>u=x^2+3</math>
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|width="100%"| <math>\displaystyle \int (x^2+3)^5x \, dx\quad</math> by using the substitution <math>u=x^2+3</math>,
|-
|-
|c)
|c)
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|width="100%"| <math>\displaystyle \int x^2 e^{x^3} \, dx\quad</math> genom att använda substitution <math>u=x^3</math>
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|width="100%"| <math>\displaystyle \int x^2 e^{x^3} \, dx\quad</math> by using the substitution <math>u=x^3</math>.
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.2:1|Lösning a|Lösning 2.2:1a|Lösning b|Lösning 2.2:1b|Lösning c|Lösning 2.2:1c}}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:1|Solution a|Solution 2.2:1a|Solution b|Solution 2.2:1b|Solution c|Solution 2.2:1c}}
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===Övning 2.2:2===
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===Exercise 2.2:2===
<div class="ovning">
<div class="ovning">
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Tolka integralerna som areor och bestäm deras värde
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Calculate the integrals
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="50%"|<math>\displaystyle\int_{-1}^{2} 2\, dx</math>
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|width="50%"|<math>\displaystyle\int_{0}^{\pi} \cos 5x\, dx</math>
|b)
|b)
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|width="50%"| <math>\displaystyle\int_{0}^{1} (2x+1)\, dx</math>
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|width="50%"| <math>\displaystyle\int_{0}^{1/2} e^{2x+3}\, dx</math>
|-
|-
|c)
|c)
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|width="50%"| <math>\displaystyle \int_{0}^{2} (3-2x)\, dx</math>
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|width="50%"| <math> \displaystyle\int_{0}^{5} \sqrt{3x + 1} \, dx</math>
|d)
|d)
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|width="50%"| <math>\displaystyle\int_{-1}^{2}|x| \, dx</math>
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|width="50%"| <math>\displaystyle\int_{0}^{1} \sqrt[\scriptstyle3]{1 - x}\, dx</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.2:2|Lösning a|Lösning 2.2:2a|Lösning b|Lösning 2.2:2b|Lösning c|Lösning 2.2:2c|Lösning d|Lösning 2.2:2d}}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:2|Solution a|Solution 2.2:2a|Solution b|Solution 2.2:2b|Solution c|Solution 2.2:2c|Solution d|Solution 2.2:2d}}
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===Exercise 2.2:3===
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<div class="ovning">
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Calculate the integrals
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%"|<math>\displaystyle\int 2x \sin x^2\, dx</math>
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|b)
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|width="50%"| <math>\displaystyle\int \sin x \cos x\, dx</math>
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|-
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|c)
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|width="50%"| <math> \displaystyle\int \displaystyle\frac{\ln x}{x}\, dx</math>
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|d)
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|width="50%"| <math>\displaystyle\int \displaystyle\frac{x+1}{x^2+2x+2}\, dx</math>
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|-
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|e)
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|width="50%"| <math> \displaystyle\int \displaystyle\frac{3x}{x^2+1}\, dx</math>
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|f)
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|width="50%"| <math>\displaystyle\int \displaystyle\frac{\sin \sqrt{x}}{\sqrt{x}}\, dx</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:3|Solution a|Solution 2.2:3a|Solution b|Solution 2.2:3b|Solution c|Solution 2.2:3c|Solution d|Solution 2.2:3d|Solution e|Solution 2.2:3e|Solution f|Solution 2.2:3f}}
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===Exercise 2.2:4===
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<div class="ovning">
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Use the formula
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<center> <math>\int \frac{dx}{x^2+1} = \arctan x + C</math> </center>
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to calculate the integrals
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{| width="100%" cellspacing="10px"
 +
|a)
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|width="50%"|<math>\displaystyle\int \frac{dx}{x^2+4}</math>
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|b)
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|width="50%"| <math>\displaystyle\int \frac{dx}{(x-1)^2+3}</math>
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|-
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|c)
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|width="50%"| <math> \displaystyle\int \frac{dx}{x^2+4x+8}</math>
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|d)
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|width="50%"| <math>\displaystyle\int \frac{x^2}{x^2 +1}\, dx</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:4|Solution a|Solution 2.2:4a|Solution b|Solution 2.2:4b|Solution c|Solution 2.2:4c|Solution d|Solution 2.2:4d}}

Current revision

       Theory          Exercises      

Exercise 2.2:1

Calculate the integrals

a) \displaystyle \displaystyle \int_{1}^{2} \displaystyle\frac{dx}{(3x-1)^4}\quad by using the substitution \displaystyle u=3x-1,
b) \displaystyle \displaystyle \int (x^2+3)^5x \, dx\quad by using the substitution \displaystyle u=x^2+3,
c) \displaystyle \displaystyle \int x^2 e^{x^3} \, dx\quad by using the substitution \displaystyle u=x^3.
Answer
Answer 2.2:1
Solution a
Solution 2.2:1a
Solution b
Solution 2.2:1b
Solution c
Solution 2.2:1c

Exercise 2.2:2

Calculate the integrals

a) \displaystyle \displaystyle\int_{0}^{\pi} \cos 5x\, dx b) \displaystyle \displaystyle\int_{0}^{1/2} e^{2x+3}\, dx
c) \displaystyle \displaystyle\int_{0}^{5} \sqrt{3x + 1} \, dx d) \displaystyle \displaystyle\int_{0}^{1} \sqrt[\scriptstyle3]{1 - x}\, dx
Answer
Answer 2.2:2
Solution a
Solution 2.2:2a
Solution b
Solution 2.2:2b
Solution c
Solution 2.2:2c
Solution d
Solution 2.2:2d

Exercise 2.2:3

Calculate the integrals

a) \displaystyle \displaystyle\int 2x \sin x^2\, dx b) \displaystyle \displaystyle\int \sin x \cos x\, dx
c) \displaystyle \displaystyle\int \displaystyle\frac{\ln x}{x}\, dx d) \displaystyle \displaystyle\int \displaystyle\frac{x+1}{x^2+2x+2}\, dx
e) \displaystyle \displaystyle\int \displaystyle\frac{3x}{x^2+1}\, dx f) \displaystyle \displaystyle\int \displaystyle\frac{\sin \sqrt{x}}{\sqrt{x}}\, dx
Answer
Answer 2.2:3
Solution a
Solution 2.2:3a
Solution b
Solution 2.2:3b
Solution c
Solution 2.2:3c
Solution d
Solution 2.2:3d
Solution e
Solution 2.2:3e
Solution f
Solution 2.2:3f

Exercise 2.2:4

Use the formula

\displaystyle \int \frac{dx}{x^2+1} = \arctan x + C

to calculate the integrals

a) \displaystyle \displaystyle\int \frac{dx}{x^2+4} b) \displaystyle \displaystyle\int \frac{dx}{(x-1)^2+3}
c) \displaystyle \displaystyle\int \frac{dx}{x^2+4x+8} d) \displaystyle \displaystyle\int \frac{x^2}{x^2 +1}\, dx
Answer
Answer 2.2:4
Solution a
Solution 2.2:4a
Solution b
Solution 2.2:4b
Solution c
Solution 2.2:4c
Solution d
Solution 2.2:4d
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