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I don't understand why $2k-1>0$ and $f$ is a positive function given in this question? How would that be of any use?

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If $2k-1<0$ in contradiction to what is given in the question, then $k<1-k$ and so you will find that in the fourth line of the solution, $$I_1=-I_2+I_1\implies I_2=0$$ and you will not get any unique answer.