Recognising antiderivatives

Idea. By the fundamental theorem of calculus, $\int_a^bf=F(b)-F(a)$ for any antiderivative $F$. The task is to *recognise* $F$, which is the reverse of differentiating a composition.

$I_1$. $\left(-\frac1t\right)'=\frac1{t^2}$, so $I_1=\left[-\frac1t\right]_1^2=-\frac12+1=\frac12$.

$I_2$. $(\ln(\ln x))'=\frac1{\ln x}\cdot\frac1x$, so $I_2=\bigl[\ln(\ln x)\bigr]_e^3=\ln(\ln3)-\ln(\ln e)=\ln(\ln3)$, because $\ln e=1$ and $\ln1=0$.

$I_3$. $\left(e^{\sqrt x}\right)'=e^{\sqrt x}\cdot\frac1{2\sqrt x}$, so $I_3=\bigl[e^{\sqrt x}\bigr]_1^2=e^{\sqrt2}-e$.

Pattern. In each case the integrand has the form $u'\,g(u)$: a chain rule read backwards.