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I've seen that (tough) integrals may be broken into answers in logarithmic form. In other words, many integrals have an alternate answer that is in the form of a function involving logarithms. An example is this question, which gives an alternate answer in terms of logarithms.

I'd like to know much more about breaking integrals into logarithms. Is there a method that can accomplish this without luck? I've read a reference (actually pictures of a book, I believe) that stated something like any integral can be broken into this logarithmic form. I'd like to know what is known about this, and I'd be delighted if someone could reference this research.

I'm looking into an algorithm to do very tough integration, and wonder if this technique is anywhere close to feasible.

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I don't know whether there is any such method or not, but here is how I look into it: $$\int \frac{f'(x)}{f(x)} \ dx = \log{f(x)} + C$$ That is you have to make sure that the numerator is the derivative of the denominator, in case you want $\log$ to come into play.

In the case of $\int\frac{1}{\sin{x} \cdot \cos{x}}$ this method gives you the answer very quickly. $$\int\frac{1}{\sin{x} \cdot \cos{x}} \ dx = \int\frac{\sec^{2}{x}}{\tan{x}} \ dx$$ by multiplying the numerator and denominator by $\sec^{2}{x}$. Note that the derivative of $\tan{x}$ is $\sec^{2}{x}$. All depends on how you manipulate and make substitutions.

Consider this example. $$\int\frac{1}{\sqrt{\sin^{3}{x} \cos{x}}} \ dx$$ again you can write this integral as $$\int \frac{\sec^{2}{x}}{\sqrt{\tan^{3}{x}}} \ dx$$ and apply the substitution $\tan{x} = \theta$. Now I am not sure whether this integral can be converted in terms of logarithms. The method which I gave is a nice example of how multiplying by a quantity in the numerator and denominator reduces a difficult looking integral into a much more simpler looking integral. Why I added this is because, when you are looking for easier methods for solving integration, the best thing is not to stick on to one method (such as converting every integral into logarithmic form).

Also calculating integral such that $\int x^{n}\cdot e^{x} \ dx$ do not involve the use of logarithms at all. The method used for evaluating such integrals, is generally using integration by parts.

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If your integrand is a rational function, there's a method that always works (at least in principle; that is, if you're able to factorize the denominator). Google for "integration partial fractions".

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