Logradæmi
#FIG 3.2
Landscape
Center
Inches
Letter
100.00
Single
-2
1200 2
2 1 0 1 -1 -1 10 0 -1 0.000 0 0 0 1 0 2
0 0 1.00 120.00 120.00
2250 3750 6987 3750
2 1 0 1 -1 -1 10 0 -1 0.000 0 0 0 1 0 2
0 0 1.00 120.00 120.00
3031 4550 3031 1350
2 1 0 1 8 -1 10 0 -1 0.000 0 0 0 0 0 82
2677 4550 2724 4455 2778 4342 2831 4224 2884 4101 2938 3975
2991 3847 3044 3718 3097 3589 3151 3461 3204 3337 3257 3216
3310 3100 3364 2990 3417 2887 3470 2790 3524 2701 3577 2619
3630 2545 3683 2478 3737 2418 3790 2366 3843 2321 3897 2282
3950 2249 4003 2221 4056 2199 4110 2182 4163 2168 4216 2159
4270 2153 4323 2150 4376 2150 4429 2153 4483 2157 4536 2164
4589 2172 4643 2182 4696 2193 4749 2204 4802 2217 4856 2231
4909 2245 4962 2260 5016 2275 5069 2290 5122 2306 5175 2322
5229 2338 5282 2354 5335 2371 5389 2387 5442 2403 5495 2419
5548 2435 5602 2450 5655 2466 5708 2482 5761 2497 5815 2512
5868 2527 5921 2542 5975 2556 6028 2570 6081 2584 6134 2598
6188 2612 6241 2625 6294 2638 6348 2651 6401 2664 6454 2677
6507 2689 6561 2701 6614 2713 6667 2724 6721 2736 6774 2747
6827 2758 6880 2769 6934 2780 6987 2790
2 1 0 1 8 7 50 0 -1 0.000 0 0 -1 0 0 2
5700 4500 5700 1650
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3150 3750 4701 2194
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3300 3750 4800 2250
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3450 3750 4950 2250
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3600 3750 5058 2289
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3000 3750 4575 2175
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3150 3450 4425 2175
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3300 3150 4275 2175
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3450 2850 4125 2175
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3750 3750 5175 2325
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
3900 3750 5284 2362
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
4050 3750 5400 2400
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
4200 3750 5517 2427
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
4350 3750 5625 2475
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
4500 3750 5700 2550
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
4650 3750 5700 2700
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
4800 3750 5700 2850
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
4950 3750 5700 3000
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
5100 3750 5700 3150
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
5250 3750 5700 3300
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
5400 3750 5700 3450
2 1 0 1 11 7 50 0 -1 0.000 0 0 -1 0 0 2
5550 3750 5700 3600
2 1 0 1 0 7 50 0 -1 0.000 0 0 -1 0 0 2
4350 3750 4350 3825
4 0 0 50 0 0 12 0.0000 4 135 90 5775 3975 2\001
4 0 0 50 0 0 12 0.0000 4 135 90 4350 3975 1\001
Finnið flatarmálið milli $y = \frac{2x}{x^2 +1}$, \quad $x$-áss, $x=0$, $x=2$:\\
$$
A = \int_0^2 \frac{2x}{x^2 +1} \,dx = \underbrace{\int_1^5 \frac{1}{u} \, du}_{u = x^2 +1 \quad du = 2x \,dx} = \ln 5 - \ln 1 = \ln 5
$$
Details
\underline{Ath:}
Getum nú heildað ýmis föll eins og t.d. $\tan x$.
\begin{eqnarray*}
\int \tan x \,dx &= \int \frac{\sin x}{\cos x} \,dx\\
&= \underbrace{-\int \frac{1}{u}\,du }_{u = \cos x, \quad -\,du = \sin x \,dx}\\
&= - \ln |u| + C\\
&= - \ln | \cos x | + C\\
(&= \ln |\sec x | + C)
\end{eqnarray*}
Examples
\begin{enumerate}
\item
$$
\int_5^{10} \frac{1}{x} \,dx = \left[ \ln |x| \right]_5^{10} = \ln 10 - \ln 5 = \ln 2
$$
\item
$$
\int \frac{1}{1-4x} \,dx = \underbrace{\int \frac{1}{u} \left( \frac{du}{-4} \right)}_{u = 1-4x, \quad du = -4\,dx} = \frac{-1}{4} \ln |u| = \frac{-1}{4} \ln |1-4x| + C
$$
\item Finnið flatarmálið milli $y = \frac{2x}{x^2 +1}$, \quad $x$-áss, $x=0$, $x=2$:
$$
A = \int_0^2 \frac{2x}{x^2 +1} \,dx = \underbrace{\int_1^5 \frac{1}{u} \, du}_{u = x^2 +1 \quad du = 2x \,dx} = \ln 5 - \ln 1 = \ln 5
$$
\end{enumerate}