integral arctan(e^cos(x))

I don't know how to do this indefinite integral, but I believe I've figured out the definite integral by exploiting this symmetry: [imath]\cos (\pi-x) = -\cos x[/imath].
 
If Wolfram Alpha is right, this has to be done as a definite integral. What techniques have you learned for finding a definite integral when you can't find an antiderivative? @blamocur's symmetry idea is a good one; you might even try sketching a graph of the integrand with that in mind.
 
its a definitive integral from 0 to pi, to be honest I just know to integrate by parts and by substitution, and those 2 are not taking me anywhere

Dr.Peterson said:
If Wolfram Alpha is right, this has to be done as a definite integral. What techniques have you learned for finding a definite integral when you can't find an antiderivative? @blamocur's symmetry idea is a good one; you might even try sketching a graph of the integrand with that in mind.
Click to expand...
 
Raphaeld36 said:
its a definitive integral from 0 to pi, to be honest I just know to integrate by parts and by substitution, and those 2 are not taking me anywhere
Click to expand...
You can't find the indefinite integral; don't try. Note that my question was, "What techniques have you learned for finding a definite integral when you can't find an antiderivative?".

The focus is on definite integrals; it's a sort of trick question, probably meant to make you think about what the area under a curve means. Possibly they will have given you an example of that which may not look related to this one, but would use similar ideas.

If you see that the integrand itself has the same kind of symmetry as the cosine, this can be easy. Thinking about what the graph will look like will help.

If I were helping you in person, I would be looking at your textbook to see what ideas it has covered that would be used here most directly; the idea is somewhat related to the idea of the mean of a function over an interval. Sometimes symmetry will be explicitly mentioned in such problems.
 
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