# Solving Impossible Integral for MS Thesis - Civil Engineering

• jrautenb
In summary, the conversation was about a specific integral related to a civil engineering thesis. The integrand involved a polynomial with constants and a second derivative. The participants offered suggestions on how to integrate the polynomial, including using an online integrator and expanding and integrating term by term. The final solution involved multiplying the polynomial with the given E(x) and integrating it like any other polynomial, with the addition of a constant at the end.
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

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.

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

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.

## 1. What is the purpose of solving an impossible integral for a MS thesis in civil engineering?

The purpose of solving an impossible integral for a MS thesis in civil engineering is to demonstrate the ability to apply advanced mathematical techniques and problem-solving skills to real-world engineering problems. It also allows for the exploration and development of new methods and approaches to solving complex integrals, which can be useful in various civil engineering applications.

## 2. What makes an integral impossible to solve?

An integral is considered impossible to solve when it cannot be expressed in terms of elementary functions, such as polynomials, trigonometric functions, or exponential functions. This often occurs when the integrand is too complex or involves special functions that do not have closed-form solutions.

## 3. How do you approach solving an impossible integral?

There is no one definitive approach to solving an impossible integral, as it depends on the specific integral and the techniques and knowledge available to the researcher. Some possible approaches include using advanced integration techniques, approximations, numerical methods, or developing new methods specifically for that integral.

## 4. What challenges may arise when solving an impossible integral for a MS thesis in civil engineering?

Solving an impossible integral can present several challenges, such as the need for advanced mathematical knowledge, the complexity of the integrand, and the potential for errors or inaccuracies in the solution. It may also require a significant amount of time and effort to develop and test new methods for solving the integral.

## 5. How can the results of solving an impossible integral be applied in civil engineering?

The results of solving an impossible integral can have various applications in civil engineering, depending on the specific problem being addressed. For example, it may be used to analyze the behavior of structures, optimize designs, or solve complex engineering problems that require advanced mathematical modeling. It can also contribute to the development of more efficient and accurate methods for solving integrals in future engineering projects.

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