The
S(x,t) = p(x,t) + k * int_0^t h(t) dt
version is at least confusing, as it is using t both as a parameter to S and as the name of the variable to integrate over in the int() part.
S(x,t) = p(x,t) + k * int_0^t h(\tau) d\tau
can be written as
S = @(x, t) p(x,t) + k * integral(@(tau) h(tau), 0, t)
and
S(x,t) = p(x,t) + k * int_0^t h(t) dt
can potentially be written as
S = @(x, t) p(x,t) + k * integral(@(t) h(t), 0, t)
which would evaluate to the same. However, consider that if someone wrote
int_0^x sin(x)
they would normally mean the indefinite integral int( sin(x) ) -- integration of sin(x) with upperbound x. That is not the same thing that is happening with
S = @(x, t) p(x,t) + k * integral(@(t) h(t), 0, t)
which is first substituting the current specific value of t into the upper bound and then happening to use a lexically-different t as the name of the anonymous variable.
