What does an integral sign with only an upper limit mean?

Question

My course notes (Mathematics BSc, second year module on differential equations, unpublished) use expressions like $$ \int^x\frac{b}{t}\text{d}t = b\ln(x). $$ Is there an implied lower limit of $0$ (edit: $1$)?

Answer 1

We know that, if a function $f(x)$ has anti-derivative $F(x)$, then: $$\int_{a}^b f(x)\ dx=F(b)-F(a)$$ The notation implies such a lower limit, $a$, has been implicitly assumed, so that $F(a)=0$. So, in your given example, implied lower limit is $t=1$. Basically they want to calculate the indefinite integral with arbitrary constant $C=0$.