# *Impossible* Integral?

1. Jan 28, 2009

### jrautenb

Hello all,

I'm working on a problem related to my MS thesis (in Civil Engineering, not mathematics). I have come across an integral on which I have spent entirely too long.

If I could get some pointers, I'd greatly appreciate it.

$$\int [E(x) \cdot (u''(x))^2 \cdot dx]$$

It is a definite integral over the length, L, of a beam, but that doesn't really matter too much.
E(x) = Eo*(1+x/L) with Eo and L being constants
u(x) = a2*x^2+a3*x^3+a4*x^4 with a1, a2, a3, and a4 being constants
u''(x) is the second derivative of u(x) with respect to x.

Any help would be greatly appreciated.

Thanks,
JRautenb

2. Jan 28, 2009

### Citan Uzuki

I'm not sure why you're having trouble here -- according to the definitions of E and u you have given, the integrand is a polynomial. Just expand the polynomial and integrate term by term, and don't worry about doing anything fancy.

3. Jan 31, 2009

### 吴黎明

just look up the wolfram integrator online, and do it there.

4. Feb 1, 2009

### Tomtom

I agree with citan. There's nothing difficult there at all.

Derivating u(x) gives $$u''(x)=12 \cdot [a4] \cdot x^2+6 \cdot [a3] \cdot x$$

$$(u''(x))^2=144 \cdot [a4]^2 \cdot x^4+144 \cdot [a4] \cdot [a3]x^2+36 \cdot [a3]^2 \cdot x^2$$

Multiply this with your E(x) (and E(x) is simplified to Eo + (Eo/L)*x).

Then you simply integrate it as you would with any polynomial. Remember adding the constant at the end of the integration.