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I'm confused with the last step of extraction MFCC where we have to take DCT or IDCT.

Cepstrum is defined as IDFT of log(power spectrum). Since MFCC are cepstral coefficients so I think we have to take IDCT at the last step.

It's confirmed by several articles. For example this one (point 2.4 we see F^(-1)) and this one (scheme on the second page). In fact in first link formula below F(^-1) is DCT-2. Why? As wiki says DCT-3 is an inverse transform.

And if we look at implementations they take DCT-2 too. I checked two of them: pyAudioAnalysis (stMFCC method) and implementation (mfcc method) by the guy who writes for practicalcryptography.com. And this is also confirmed by several links: here and here (second page).

Well.. who is right?

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I don't believe there is a right answer. DCT or IDCT will achieve the same purpose: decorrelating to put most energy in fewer coefficients. Whatever you do next with your MFCC (compression, feature extraction) will work with either approach.

Similarly, I think there are implementations that do an FFT instead of an IFFT for the first stage of the MFCC extraction.

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