2.2 Übungen
Aus Online Mathematik Brückenkurs 2
(Unterschied zwischen Versionen)
(Translated links into English) |
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===Exercise 2.2:1=== | ===Exercise 2.2:1=== | ||
<div class="ovning"> | <div class="ovning"> | ||
| - | Calculate the | + | Calculate the integrals |
{| 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> by using the substitution <math>u=3x-1</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> by using the substitution <math>u=x^2+3</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> by using the substitution <math>u=x^3</math> | + | |width="100%"| <math>\displaystyle \int x^2 e^{x^3} \, dx\quad</math> by using the substitution <math>u=x^3</math>. |
|} | |} | ||
| - | </div>{{#NAVCONTENT: | + | </div>{{#NAVCONTENT:Answer|Svar 2.2:1|Solution a|Lösning 2.2:1a|Solution b|Lösning 2.2:1b|Solution c|Lösning 2.2:1c}} |
===Exercise 2.2:2=== | ===Exercise 2.2:2=== | ||
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|width="50%"| <math>\displaystyle\int_{0}^{1} \sqrt[\scriptstyle3]{1 - x}\, dx</math> | |width="50%"| <math>\displaystyle\int_{0}^{1} \sqrt[\scriptstyle3]{1 - x}\, dx</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT: | + | </div>{{#NAVCONTENT:Answer|Svar 2.2:2|Solution a|Lösning 2.2:2a|Solution b|Lösning 2.2:2b|Solution c|Lösning 2.2:2c|Solution d|Lösning 2.2:2d}} |
===Exercise 2.2:3=== | ===Exercise 2.2:3=== | ||
| Zeile 57: | Zeile 57: | ||
|width="50%"| <math>\displaystyle\int \displaystyle\frac{\sin \sqrt{x}}{\sqrt{x}}\, dx</math> | |width="50%"| <math>\displaystyle\int \displaystyle\frac{\sin \sqrt{x}}{\sqrt{x}}\, dx</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT: | + | </div>{{#NAVCONTENT:Answer|Svar 2.2:3|Solution a|Lösning 2.2:3a|Solution b|Lösning 2.2:3b|Solution c|Lösning 2.2:3c|Solution d|Lösning 2.2:3d|Solution e|Lösning 2.2:3e|Solution f|Lösning 2.2:3f}} |
===Exercise 2.2:4=== | ===Exercise 2.2:4=== | ||
| Zeile 63: | Zeile 63: | ||
Use the formula | 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 | + | to calculate the integrals |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 75: | Zeile 75: | ||
|width="50%"| <math>\displaystyle\int \frac{x^2}{x^2 +1}\, dx</math> | |width="50%"| <math>\displaystyle\int \frac{x^2}{x^2 +1}\, dx</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT: | + | </div>{{#NAVCONTENT:Answer|Svar 2.2:4|Solution a|Lösning 2.2:4a|Solution b|Lösning 2.2:4b|Solution c|Lösning 2.2:4c|Solution d|Lösning 2.2:4d}} |
Version vom 11:42, 3. Sep. 2008
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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. |
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 |
Answer
Solution a
Solution b
Solution c
Solution 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 |
Answer
Solution a
Solution b
Solution c
Solution d
Solution e
Solution f
Exercise 2.2:4
Use the formula
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
Solution a
Solution b
Solution c
Solution d
