Integrating 1/[(sin^3x)+(cos^3x)] dx: Step-by-Step Solution

[ardie]

well the thing is the denominator has so many essential singularities that the integral seems almost useless in mathematical applications
did you really have to do it before you could get some sleep? :D

 

[SammyS]

After looking at a graph of the integrand, it occurred to me to look at the following:

This seemingly simple substitution may help.

Let u = x + π/4

\displaystyle \frac{1}{\sin^3(u-\pi/4)+\cos^3(u-\pi/4)}

\displaystyle =\frac{2\sqrt{2}}{3\sin(u)+\sin(3u)}

ehild,

This is pretty embarrassing! When I was putting together the above quoted post, I looked through the previous posts, thinking that I might have seen a similar substitution, u = x + π/4, particularly in one of your posts.

I guess I was pretty tired and/or had a brain cramp, but I missed the fact that this was the initial substitution you made in your very nice solution. When going over your solution to try to check it, I graphed \displaystyle \frac{\sqrt{2}}{ sin(u)(1+2 cos^2(u))}\,, and saw that it was the same graph as I had obtained for the above expressions.

I would have never posted that final expression -- the one with the sin(3u) .

At any rate, your solution looks good. I have no idea how you got WolframAlpha to get that rather uncomplicated answer. Both my expressions gave a much more complicated answer.

 

[ehild]

well the thing is the denominator has so many essential singularities that the integral seems almost useless in mathematical applications
did you really have to do it before you could get some sleep? :D

Yes, the graph of the integrand looks terrible, and the whole thing has not much sense, but it was challenging. And the next problem could be to find the improper integral between -pi/4 and 3pi/4.
I could not sleep and when I fall asleep at last, I dreamt of the integral and awoke as I recognised (in my dream) that impossible to derive it without pen an paper.

ehild

 

[ehild]

I have no idea how you got WolframAlpha to get that rather uncomplicated answer. Both my expressions gave a much more complicated answer.

It was the first time that I found and used Wolframalpha. I just typed in integral(1/((sin(x))^3+(cos(x))^3))dx, and it gave the expression with i, but scrolling down, there were alternate forms, one of them that relatively simple one, without i.

ehild