= First integral of the Chebyshev function
{title2=$\Psi_1(x)=\int_0^x\psi(u)\,du$}
For the <Second Chebyshev function>, finite summation gives $\Psi_1(x)=\sum_{n\le x}(x-n)\Lambda(n)$. For $c>1$, Mellin inversion of the kernel $1/(s(s+1))$ gives
$$
\Psi_1(x)=\frac1{2\pi i}\int_{c-i\infty}^{c+i\infty}-\frac{\zeta'(s)}{\zeta(s)}\frac{x^{s+1}}{s(s+1)}\,ds.
$$
The extra denominator makes this smoothing useful: the resulting sum over <Nontrivial zeros of the Riemann zeta function> is absolutely convergent. This integral differs from the <integrated Chebyshev sum>, which denotes a sum of the values $\psi(x/n)$.
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