2.2 Övningar
Sommarmatte 2
(Skillnad mellan versioner)
| Versionen från 5 juni 2007 kl. 08.13 (redigera) KTH.SE:u1zpa8nw (Diskussion | bidrag) (→Övning 2.2:4) ← Gå till föregående ändring |
Versionen från 17 juni 2007 kl. 14.59 (redigera) (ogör) KTH.SE:u1tyze7e (Diskussion | bidrag) (Korrekturläst) Gå till nästa ändring → |
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| Rad 30: | Rad 30: | ||
| <tr align="left" valign="top"> | <tr align="left" valign="top"> | ||
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="100%">$\displaystyle\frac{x^{10}}{12}+C$</td> | + | <td class="ntext" width="100%">$\displaystyle\frac{(x^2+3)^6}{12}+C$</td> |
| </tr> | </tr> | ||
| <tr align="left" valign="top"> | <tr align="left" valign="top"> | ||
| Rad 112: | Rad 112: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="50%">$\cos x^2+C$</td> | + | <td class="ntext" width="50%">$-\cos x^2+C$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle\frac{\cos2x}{4}+C$</td> | + | <td class="ntext" width="50%">$\displaystyle\frac{\sin^2x}{2}+C$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$(\ln x)^2$</td> | + | <td class="ntext" width="50%">$\frac{1}{2}(\ln x)^2+C$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| <td class="ntext" width="50%">$\displaystyle\frac{1}{2}\ln\left(x^2+2x+2\right)+C$</td> | <td class="ntext" width="50%">$\displaystyle\frac{1}{2}\ln\left(x^2+2x+2\right)+C$</td> | ||
| Rad 137: | Rad 137: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle\int \displaystyle\frac{dx}{x^2+4}$</td> | + | <td class="ntext" width="50%">$\displaystyle\int \frac{dx}{x^2+4}$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle\int \displaystyle\frac{dx}{(x-1)^2+3}$</td> | + | <td class="ntext" width="50%">$\displaystyle\int \frac{dx}{(x-1)^2+3}$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$ \displaystyle\int e\displaystyle\frac{dx}{x^2+4x+8}$</td> | + | <td class="ntext" width="50%">$ \displaystyle\int \frac{dx}{x^2+4x+8}$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle\int \displaystyle\frac{x^2}{x^2 +1}\, dx$</td> | + | <td class="ntext" width="50%">$\displaystyle\int \frac{x^2}{x^2 +1}\, dx$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
Versionen från 17 juni 2007 kl. 14.59
Innehåll |
Övning 2.2:1
Beräkna integralerna
| a) | $\displaystyle \int_{1}^{2} \displaystyle\frac{dx}{(3x-1)^4}$ genom att använda substitution $u=3x-1$ |
| b) | $\displaystyle \int (x^2+3)^5x \, dx$ genom att använda substitution $u=x^2+3$ |
| c) | $\displaystyle \int x^2 e^{x^3} \, dx$ genom att använda substitution $u=x^3$ |
Facit
Facit till alla delfrågorna
| a) | $\displaystyle\frac{13}{1000}$ |
| b) | $\displaystyle\frac{(x^2+3)^6}{12}+C$ |
| c) | $\displaystyle\frac{1}{3}e^{\scriptstyle x^3}+C$ |
Övning 2.2:2
Beräkna integralerna
| a) | $\displaystyle\int_{0}^{\pi} \cos 5x\, dx$ | b) | $\displaystyle\int_{0}^{1/2} e^{2x+3}\, dx$ |
| c) | $ \displaystyle\int_{0}^{5} \sqrt{3x + 1} \, dx$ | d) | $\displaystyle\int_{0}^{1} \sqrt[\scriptstyle3]{1 - x}\, dx$ |
Facit
Facit till alla delfrågorna
| a) | $0$ | b) | $\displaystyle\frac{1}{2}(e^4-e^3)$ |
| c) | $14$ | d) | $\displaystyle\frac{3}{4}$ |
Övning 2.2:3
Beräkna integralerna
| a) | $\displaystyle\int 2x \sin x^2\, dx$ | b) | $\displaystyle\int \sin x \cos x\, dx$ |
| c) | $ \displaystyle\int \displaystyle\frac{\ln x}{x}\, dx$ | d) | $\displaystyle\int \displaystyle\frac{x+1}{x^2+2x+2}\, dx$ |
| e) | $ \displaystyle\int \displaystyle\frac{x}{x^2+1}\, dx$ | f) | $\displaystyle\int \displaystyle\frac{\sin \sqrt{x}}{\sqrt{x}}\, dx$ |
Facit
Facit till alla delfrågorna
| a) | $-\cos x^2+C$ | b) | $\displaystyle\frac{\sin^2x}{2}+C$ |
| c) | $\frac{1}{2}(\ln x)^2+C$ | d) | $\displaystyle\frac{1}{2}\ln\left(x^2+2x+2\right)+C$ |
| e) | $\displaystyle\frac{1}{2}\ln\left(x^2+1\right)+C$ | f) | $-2\cos\sqrt{x}+C$ |
Övning 2.2:4
Använd formeln $$\int \frac{dx}{x^2+1} = \arctan x + C$$ för att beräkna integralerna
| a) | $\displaystyle\int \frac{dx}{x^2+4}$ | b) | $\displaystyle\int \frac{dx}{(x-1)^2+3}$ |
| c) | $ \displaystyle\int \frac{dx}{x^2+4x+8}$ | d) | $\displaystyle\int \frac{x^2}{x^2 +1}\, dx$ |
Facit
Facit till alla delfrågorna
| a) | $\displaystyle\frac{1}{2}\arctan\left(\frac{x}{2}\right)+C$ | b) | $\displaystyle\frac{1}{\sqrt3}\arctan\left(\frac{x-1}{\sqrt3}\right)+C$ |
| c) | $\displaystyle\frac{1}{2}\arctan\left(\frac{x+2}{2}\right)+C$ | d) | $x-\arctan x + C$ |
