Theoretical Integral Question. Please help.?

Question

Can someone explain why this integral cannot be done:

sin(x)/ln(x) dx

Even the wolfram online integrator cannot do it. The curve is continuous completely, so i don't quite understand why it cannot be possibly done.

It looks like this:

Link: http://www.btlstudios.com/uploads/files/kk_png305945.png

I hear people talk about how functions become too complicated to integrate, when the representation of the new function isn't in a basic form or whatever.

Please explain.

Answer 1

Firstly, being continuous doesn't mean it is integrable.

Secondly, when we say it is no integrable, we mean that there is no closed form solution, i.e. we cannot express the answer in radicals, trigo, exponential, logarithm.

Thirdly, many functions in real life application do not have closed form solutions and can only be solved using numerical methods.