# A Integral calculation (Most Difficult Integral)

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1. Mar 2, 2016

### FL0R1

Can anybody give me a solution for the integral of [(lnx)/(cosx)]dx??
I would be very gratefully even they do go in a way

2. Mar 2, 2016

### zinq

I very much doubt that the expression ln(x) / cos(x) is the antiderivative of any function that can be expressed in closed form in terms of elementary functions.

Of course, any well-defined differentiable function does have an indefinite integral, but in some sense *most* cases this antiderivative cannot be expressed in any simple closed form (meaning with a finite number of elementary functions as terms).

On the other hand, if the pieces of the original function can be expressed as power series that converge in the same region — as is the case for ln(x) and cos(x) — then there is a power series expression for the antiderivative, at least where there is no zero in the denominator. In the present case, ln(x) is not defined at x = 0, but we can find a power series expansion for the antiderivative of ln(x) / cos(x) about the point x = 1.

It is likely to be have complicated coefficients, but a computer algebra system like Mathematica can in principle calculate any number of terms of the series.

3. Mar 3, 2016

### FL0R1

I putted this in Mathematica program and it didnt gave results, when i put it the program stopped, anyway.This is a indefinite integral and sure that can be found by power series but do anybody know any algebra "trick" that can go to a way.
Thanks

4. Mar 3, 2016

### dextercioby

First, asking $\int \frac{\ln x}{\cos x} {}dx = ?$ is not a properly formulated mathematical problem. You need to specify for which values of x the function under the integral sign makes sense in the reals, or in the complex plane. Only then you can ask yourself, for what values of x, does it make sense to consider a series expansion...

5. Mar 4, 2016

### HallsofIvy

Staff Emeritus
Do you understand that "almost all" integrals of elementary functions cannot be written in terms of elementary functions?

6. Mar 4, 2016

### FL0R1

its an undefinite integral..

7. Mar 4, 2016

### FL0R1

Im in high school and we didnt still learned power series, i just wanted a solution for my grade.Sure that i know power series but if i solve it with PS how can classmates understand.

8. Mar 4, 2016

### Staff: Mentor

There is no solution for high school.