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In[]:=

initialExpr[u0_,x_,t_,k_]:=u0*(1-(1/Pi)*Integrate[Exp[-v*t]/v*Sin[Sqrt[v/k]*x],{v,0,Infinity},Assumptions->{t>0,k>0}])

In[]:=

eulerSubstitute=Sin[Sqrt[v/k]*x]->(Exp[I*Sqrt[v/k]*x]-Exp[-I*Sqrt[v/k]*x])/(2*I);

In[]:=

modifiedExpr=initialExpr[u0,x,t,k]/.eulerSubstitute

Out[]=

u01-Erf

x

2

kt

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