2.2 Exercises
From Förberedande kurs i matematik 2
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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>. | |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:Answer|Answer 2.2:1|Solution a| | + | </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}} |
===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> | ||
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| - | </div>{{#NAVCONTENT:Answer|Answer 2.2:2|Solution a| | + | </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}} |
===Exercise 2.2:3=== | ===Exercise 2.2:3=== | ||
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|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> | ||
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| - | </div>{{#NAVCONTENT:Answer|Answer 2.2:3|Solution a| | + | </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}} |
===Exercise 2.2:4=== | ===Exercise 2.2:4=== | ||
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|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> | ||
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| - | </div>{{#NAVCONTENT:Answer|Answer 2.2:4|Solution a| | + | </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}} |
Revision as of 07:30, 17 September 2008
| 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
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Solution a
Solution b
Solution c
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
