Cutkosky representation and direct integration

We present a new method of direct integration of Feynman integrals based on the Cutkosky representation of the integrals. In this representation we are able to explicitly compute the integrals which yield square root singularities and leave only the integrals which yield logarithmic singularities, thus making the transcendentality weight manifest. The method is elementary, algorithmic, does not introduce spurious non-physical singularities and does not require a reduction to a basis of pure integrals.