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We compute the homotopy groups of $\mathrm{THH}(\mathrm{ku})$ as a$\mathrm{ku}_\ast$-module using the descent spectral sequence for the map$\mathrm{THH}(\mathrm{ku})\to\mathrm{THH}(\mathrm{ku}/\mathrm{MU})$, which isthe motivic spectral sequence for $\mathrm{THH}(\mathrm{ku})$ in the sense ofHahn-Raksit-Wilson. We compute the $E_2$-page using an algebraic Bocksteinspectral sequence, which is an approximation to the topological Bocksteinspectral sequence computing $\mathrm{THH}(\mathrm{ku})$ from$\mathrm{THH}(\mathrm{ku})/(p,\beta)$, where $\beta$ is the Bott element. Then,we show that the descent spectral sequence degenerates at the $E_2$-page.

Integral topological Hochschild homology of connective complex K-theory