1. ## Messy Integral

In the course of solving a second order differential equation I came across this integral. I'm not sure if it's solvable or not, but so far I've had no luck.

$\displaystyle \int \frac {x * cos(x)}{1+Cx^2} dx$

I've tried substituting with $\displaystyle u = Arctan(\sqrt(C)*x)$ and with $\displaystyle u = 1 + Cx^2$ but in both cases the cosine on top made it almost as complicated as it started.

2. Originally Posted by tbgh
In the course of solving a second order differential equation I came across this integral. I'm not sure if it's solvable or not, but so far I've had no luck.

$\displaystyle \int \frac {x * cos(x)}{1+Cx^2} dx$

I've tried substituting with $\displaystyle u = Arctan(\sqrt(C)*x)$ and with $\displaystyle u = 1 + Cx^2$ but in both cases the cosine on top made it almost as complicated as it started.
There is no answer in terms of a finite number of elementary functions.

Perhaps you should post the DE you're trying to solve.

3. This has to do with the motion of a board rocking on a circular pole. What made it so complicated is as the board rocks, the point of contact changes and that changes the moment of inertia for the rocking board as well as net torque. Variables are as follows.

$\displaystyle y=$angle between the board and horizontal (I'm using clockwise as positive, not that it matters)
$\displaystyle r=$radius of pole
$\displaystyle g=$accelerational constant of gravity
$\displaystyle l=$length of the board
$\displaystyle A=$angular acceleration

Once I divided the equation for torque by the moment of inertia the mass canceled out, leaving me with this equation.

$\displaystyle A=\frac{24rgy*cos(y)}{l^2 + 4r^2y^2}$

From there eliminating the constants as best I could gave me what I put in my first post.

4. Originally Posted by tbgh
This has to do with the motion of a board rocking on a circular pole. What made it so complicated is as the board rocks, the point of contact changes and that changes the moment of inertia for the rocking board as well as net torque. Variables are as follows.

$\displaystyle y=$angle between the board and horizontal (I'm using clockwise as positive, not that it matters)
$\displaystyle r=$radius of pole
$\displaystyle g=$accelerational constant of gravity
$\displaystyle l=$length of the board
$\displaystyle A=$angular acceleration

Once I divided the equation for torque by the moment of inertia the mass canceled out, leaving me with this equation.

$\displaystyle A=\frac{24rgy*cos(y)}{l^2 + 4r^2y^2}$

From there eliminating the constants as best I could gave me what I put in my first post.
It comes as a shock to many students when they discover that most differential equations do not not have exact solutions ....

5. Any methods you would reccomend for getting the best approximate equation? I'm six years out of college so differential equations has gotten a bit fuzzy.