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$$\int \frac{x + \sin x}{\sqrt {\tan x } + \cos x} dx$$

I tried using substitution and rationalize but nothing works. And I do not how to solve this one. Please help me.

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  • $\begingroup$ is it a definite integral? Where are the limits $\endgroup$ Apr 16, 2017 at 10:39
  • $\begingroup$ Sorry my mistake. It's indefinite. $\endgroup$
    – Amar
    Apr 16, 2017 at 10:41
  • $\begingroup$ Hint: $\int\dfrac{dx}{\sqrt{\tan x}+\cos x}$ integral-calculator.com/… $\endgroup$ Apr 17, 2017 at 15:42
  • $\begingroup$ @Amar are you sure it is doable? $\endgroup$
    – Sid
    Apr 20, 2017 at 11:34
  • $\begingroup$ or is this a troll xD? $\endgroup$
    – Sid
    Apr 20, 2017 at 11:36

1 Answer 1

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Analysis using known programs indicates that the given integral does not have a closed representation through known functions. In addition, for negative tangent values ​​in the denominator, the integrand function ceases to be valid, and the statement of the problem does not contain recommendations for this case.

On the other hand, the function in the denominator is periodic.

Based on these considerations, the following should be done:

1) abandon the search for solutions in elementary functions;
2) explicitly identify the intervals where the function is valid;
3) replace the denominator by a factor in the form of an infinite Fourier series.

Under these conditions, you can get an acceptable result.

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