Reading the defintion of the IntegralCosinus $$ {\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt $$ I wonder what happens, if I to split the function in the integral: $$ \begin{eqnarray*} {\rm Ci}(x) &=& \gamma + \ln x + \int_0^x\frac{\cos t}{t}\,dt -\int_0^x\frac{1}{t}\,dt \\ &=& \gamma + \ln x + \int_0^x\frac{\cos t}{t}\,dt -\left[ \ln (t) \right]_0^x \\ &=& \gamma + \ln x + \int_0^x\frac{\cos t}{t}\,dt -\ln(x) + \underbrace{\ln(0)}_? \tag{1} \\ \end{eqnarray*} $$ Is splitting not allowed here or how do I have to interprete $\ln(0)$?
And further if I look at another definition $$ -{\rm Ci}(x) = \int_x^\infty\frac{\cos t}{t}\,dt \tag{2} $$ and now add $(1)$ and $(2)$ I get: $$ 0=\gamma + \int_0^\infty\frac{\cos t}{t}\,dt $$ This doesn't seem right. Can anybody tell me what's wrong here?