help with those questions??
The first integral is,
$\displaystyle \int_0^{\pi} \frac{x\sin x}{1+\cos ^2 x}dx$
The function is continous on $\displaystyle [0,\pi]$.
Thus we need to find,
$\displaystyle \int \frac{x\sin x}{1+\cos^2 x}dx$
Express as,
$\displaystyle \int x\cdot \frac{\sin x}{1+\cos^2 x}dx$
Let,
$\displaystyle u=x$ und $\displaystyle v'=\frac{\sin x}{1+\cos^2 x}$.
Thus,
$\displaystyle u'=1$ and $\displaystyle v=-\tan^{-1} (\cos x)$
Thus, by parts,
$\displaystyle -x\tan^{-1}(\cos x)+\int \tan^{-1}(\cos x)dx$
The nice thing is that though we do not know what that integral is we know that it is zero.
Thus,
$\displaystyle -x\tan^{-1}(\cos x)\big|^{\pi}_0 +\int_0^{\pi}\tan^{-1}(\cos x)dx$
As mentioned that integral "dies out".
$\displaystyle -\pi \tan^{-1}(-1)=-\frac{3\pi^2}{4}$
The third problem.
You have,
$\displaystyle \frac{x}{(x+1)^2(x^2+1)}$
The Partial Fractions Decomposition is,
$\displaystyle \frac{A}{x+1}+\frac{B}{x+1}+\frac{Cx+D}{x^2+1}$
With some work we find,
$\displaystyle \frac{1}{2} \left( \frac{1}{x^2+1} - \frac{1}{(x+1)^2} \right) $
Now you can integrate.
The first is acrtangent.
The second is basic substitution.
The Last one.
$\displaystyle \int_0^3 x-[x]+1/3 dx$
If you think about it,
$\displaystyle \int_0^n [x]dx$
Where $\displaystyle n$ is an integer.
Is, $\displaystyle 1+2+..+n-1=\frac{n(n-1)}{2}$
Because we can think of this as adding rectangles of area each with increasing area of 1. Thus it is an arithmetical sum.
Thus,
$\displaystyle \int_0^3 xdx -\int_0^3 [x] dx+\int_0^3 1/3 dx$
You should be able to do these now.
This is mine 39th Post!!!
This may be cheating, but I used Excel to create a midpoint Riemann Sum with n=50.
That gives $\displaystyle {\Delta}x=\frac{\pi}{50}$
Some of the numbers are skewed in the display, but that doesn't matter.
It turned out that way.
Add up the rightmost column and multiply by Pi/50 and you getCode:1 0.031416 0.000494 2 0.094248 0.004454 3 0.15708 0.012439 4 0.219912 0.024571 5 0.282743 0.041039 6 0.345575 0.062092 7 0.408407 0.088042 8 0.471239 0.119259 9 0.534071 0.156165 10 0.596903 0.199226 11 0.659735 0.248935 12 0.722566 0.305786 13 0.785398 0.37024 14 0.84823 0.442672 15 0.911062 0.5233 16 0.973894 0.612103 17 1.036726 0.70871 18 1.099558 0.812293 19 1.162389 0.921451 20 1.225221 1.034128 21 1.288053 1.147586 22 1.350885 1.258465 23 1.413717 1.362958 24 1.476549 1.457091 25 1.539381 1.537104 26 1.602212 1.599843 27 1.665044 1.643103 28 1.727876 1.665837 29 1.790708 1.668198 30 1.85354 1.651404 31 1.916372 1.617482 32 1.979204 1.568957 33 2.042035 1.508544 34 2.104867 1.438896 35 2.167699 1.362421 36 2.230531 1.281182 37 2.293363 1.196853 38 2.356195 1.11072 39 2.419027 1.023718 40 2.481858 0.936468 41 2.54469 0.849332 42 2.607522 0.762453 43 2.670354 0.675801 44 2.733186 0.589206 45 2.796018 0.502381 46 2.85885 0.414945 47 2.921681 0.326439 48 2.984513 0.236332 49 3.047345 0.144028 50 3.110177 0.04887
2.467659
Not too bad. If you run this through Maple or a TI-89 or something, you get
2.464010027.
Click on the links below to see the Riemann sum graphs of #1 and #2: