The phrase 'indispensable task' is indeed correct and can be used in written English.
You can use it when you want to emphasize a task's importance or necessity. For example: "Completing this project is an indispensable task that must be done by the end of the month."
Determining the tunnel convergence is an indispensable task in tunneling, especially when adopting the New Austrian Tunneling Method.
By the time she delivered her commencement address at Wellesley in 1969, she was citing her generation's "indispensable task of criticizing and constructive protest".
Therefore, the approach to correctly and efficiently solve the singularity arising from degenerate scale becomes a very essential and indispensable task for engineers.
Thus, constructing an ad hoc teaching model that includes didactic strategies specifically designed for the forensic sciences is an indispensable task.
Therefore, in order to perform reliable and consistent assessment on the health condition of civil engineering infrastructures, it is an indispensable task to quantify the modal frequency fluctuation induced by change of ambient conditions, such as temperature and humidity.
In-vessel operation/inspection is an indispensable task for Tokamak experimental reactor, for a robot/manipulator is more capable in doing this than human being with more precise motion and less risk of damaging the ambient equipment.
The separation and quantitation of the enantiomers and also the determination of the enantiomeric purity are now current and indispensable tasks for the pharmaceutical analysis.
Eating food and verbalizing thoughts are just a couple of the indispensable tasks a mouth can do.
Exploration of available nest sites may indeed be time consuming and require energy, but it can be carried out with little extra cost by the same individuals that go out of the nest on a daily basis to explore and perform indispensable tasks such as foraging and patrolling.
First, the National Chagas Programme, with collaboration of the National Chagas Laboratory, Medical Entomology Unit and aid agencies, drafted a provisional surveillance guideline outlining indispensable tasks at the national, departmental and local levels.
The surveillance system performance index is a checklist of 48 items composed of indispensable tasks of the National Programmes (11 items), Departmental Health Offices (12 items), health centres (14 items) and communities (11 items) [ 20]: it was designed by the bilateral project team and based on the provisional surveillance guidelines to analyse the task completion and pitfalls.
