2.2 Übungen
Aus Online Mathematik Brückenkurs 2
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| - | {{Ej vald flik|[[2.2 Variabelsubstitution| | + | {{Ej vald flik|[[2.2 Variabelsubstitution|Theory]]}} |
| - | {{Vald flik|[[2.2 Övningar| | + | {{Vald flik|[[2.2 Övningar|Exercises]]}} |
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| - | === | + | ===Exercise 2.2:1=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Calculate the integral | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| - | |width="100%"|<math>\displaystyle \int_{1}^{2} \displaystyle\frac{dx}{(3x-1)^4}\quad</math> | + | |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) | ||
| - | |width="100%"| <math>\displaystyle \int (x^2+3)^5x \, dx\quad</math> | + | |width="100%"| <math>\displaystyle \int (x^2+3)^5x \, dx\quad</math> by using the substitution <math>u=x^2+3</math> |
|- | |- | ||
|c) | |c) | ||
| - | |width="100%"| <math>\displaystyle \int x^2 e^{x^3} \, dx\quad</math> | + | |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}} | </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}} | ||
| - | === | + | ===Exercise 2.2:2=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Calculate the integrals | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 38: | Zeile 38: | ||
</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}} | </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}} | ||
| - | === | + | ===Exercise 2.2:3=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Calculate the integrals | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 59: | Zeile 59: | ||
</div>{{#NAVCONTENT:Svar|Svar 2.2:3|Lösning a|Lösning 2.2:3a|Lösning b|Lösning 2.2:3b|Lösning c|Lösning 2.2:3c|Lösning d|Lösning 2.2:3d|Lösning e|Lösning 2.2:3e|Lösning f|Lösning 2.2:3f}} | </div>{{#NAVCONTENT:Svar|Svar 2.2:3|Lösning a|Lösning 2.2:3a|Lösning b|Lösning 2.2:3b|Lösning c|Lösning 2.2:3c|Lösning d|Lösning 2.2:3d|Lösning e|Lösning 2.2:3e|Lösning f|Lösning 2.2:3f}} | ||
| - | === | + | ===Exercise 2.2:4=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Use the formula | |
<center> <math>\int \frac{dx}{x^2+1} = \arctan x + C</math> </center> | <center> <math>\int \frac{dx}{x^2+1} = \arctan x + C</math> </center> | ||
| - | + | to calculate the integral | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
Version vom 10:45, 4. Aug. 2008
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Exercise 2.2:1
Calculate the integral
| 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 |
Laden...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 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
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 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
Lösning f
Exercise 2.2:4
Use the formula
to calculate the integral
| 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 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
