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My course book bluntly mentions (freely translation without any proof):

Integral functions with the terms $x^{\alpha} \sin(\beta x)$, $x^{\alpha} \cos(\beta x)$ or $x^{\alpha}e^{\beta x}$ ($\alpha, \beta\in \mathbb R$) are elementary if $\beta=0$ or $\alpha\in \mathbb N\cup{0}$.

Unfortunately, I cannot express the function $\int \cos(x) \ln(x) dx$ in any of the forms -- I always get three terms. Is there some elegant way to know whether some function is elementary, not just looking at some constants of certain functions? Could someone explain why the functions in the forms are elementary by which theorems?

References

  1. I am doing the book alone here, ex. 5 on page 529 for future readers (sorry not English book).
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up vote 2 down vote accepted

Concerning the last question, if $\beta=0$ then the integrand is just $x^{\alpha}$, which I trust you can integrate. Similarly, if $\alpha=0$ then the integrations are not hard. If $\alpha$ is a positive integer, then you can use integration by parts to reduce the exponent on $x$ by one; repeated application brings the exponent down to zero, and the previous sentence applies.

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...I see now, just that easy -- repeated use of integration by parts. Thank you. (I would still like to know whether there is some generic definition for elementary function or some generic way to find out whethere something is elementary, there must be more into this issue.) –  hhh Jan 29 '12 at 1:44
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Yes, there is a whole theory of integration in closed form, and there are algorithms for deciding whether an integral is elementary, and for evaluating it if it is. It's too long to explain on an m.se page, but en.wikipedia.org/wiki/Risch_algorithm might get you started. –  Gerry Myerson Jan 29 '12 at 2:09
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Integrating by parts: $$ \int \cos(x) \ln(x) \mathrm{d} x = \int \ln(x) \mathrm{d} (\sin(x)) = \sin(x) \ln(x) - \int \frac{\sin(x)}{x} \mathrm{d} x $$ The integral $\int \frac{\sin(x)}{x} \mathrm{d} x$ is known to be non-elementary.

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What is it called then when $\int \cos(x) dx$, this is elementary but it cannot be called cosine integral because all cosine intergrals are non-elementary? Or do I infer it right? (They use terms such as exponential|logaritmic|etc integral with things of the form $\int \frac{func(x)}{t} dx$ and do they always mean non-elementary function?) –  hhh Jan 29 '12 at 1:36
    
"cosine integral" is a term for $\int(\cos x/x)\,dx$. –  Gerry Myerson Jan 29 '12 at 1:40
    
@GerryMyerson: sorry? They have always divisor there like here? What is $\int cos(x) dx$ called? It is elementary but is it cosine integral? Wikipedia uses the term only apparently for non-elementary functions. –  hhh Jan 29 '12 at 1:42
    
@hhh, I would call $\int \cos(x) dx$ an integral of cosine, or better yet, $\sin(x) + C$. –  Shaun Ault Jan 29 '12 at 1:49
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There are a few non-elementary integrals that are of sufficient importance that they have been given their own names. For another example, the logarithmic integral is the name for $\int dx/\log x$. Elementary integrals generally don't have names - they don't need names - you just do them. –  Gerry Myerson Jan 29 '12 at 2:05
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