Integrate(ODE, y(x)) applies an integral to both sides of an exact ODE, and then performs the integration, thereby obtaining a solution to the ODE.

Integrate:-Apply(ODE, y(x)) applies an integral to both sides of an exact ODE, without performing the integration.

Integrate:-Evaluate(ODE) tries to evaluate any unevaluated integrals in the expression ODE, if possible.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\right)\:$

$\mathrm{ode1}≔\frac{\mathrm{diff}\left(z\left(t\right)\,t\right)}{z\left(t\right)+1}{z\left(t\right)}^2=-\frac{t^2}{t-1}$

$\mathrm{ode1}≔\frac{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right){z\left(t\right)}^2}{z\left(t\right)+1}=-\frac{t^2}{t-1}$

$\mathrm{Integrate}\left(\mathrm{ode1}\,z\left(t\right)\right)$

$\frac{{z\left(t\right)}^2}{2}-z\left(t\right)+\ln \left(z\left(t\right)+1\right)=-\frac{t^2}{2}-t-\ln \left(t-1\right)+\mathrm{\_C1}$

$\mathrm{eq2}≔\mathrm{Integrate}:-\mathrm{Apply}\left(\mathrm{ode1}\,z\left(t\right)\right)$

$\mathrm{eq2}≔\int \frac{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right){z\left(t\right)}^2}{z\left(t\right)+1}\ⅆt=\int -\frac{t^2}{t-1}\ⅆt+\mathrm{\_C1}$

$\mathrm{Integrate}:-\mathrm{Evaluate}\left(\mathrm{eq2}\right)$

$\frac{{z\left(t\right)}^2}{2}-z\left(t\right)+\ln \left(z\left(t\right)+1\right)=-\frac{t^2}{2}-t-\ln \left(t-1\right)+\mathrm{\_C1}$

The Student[ODEs][Integrate] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.