Calculus with Inverse Trig Functions

Basic Concepts

Understand the inverse trigonometric functions Sine, Cosine, and Tangent.
Inverse functions written as (sin^-1(x), cos^-1(x), tan^-1(x)) or (arcsin(x), arccos(x), arctan(x)) are the results of reversing the sine, cosine and tangent functions respectively.
Domain of arcsin(x) and arccos(x) is [-1,1], and range is from [-π/2, π/2] for arcsin(x), and [0, π] for arccos(x).
Domain of arctan(x) is all real numbers, range is (-π/2, π/2).
Understand periodicity and symmetry inherent in inverse trigonometric functions.
Remember that inverse trig functions are continuous and neighborhood preserving.

The derivatives of the inverse sine (arcsin), inverse cosine (arccos) and inverse tangent (arctan) functions are crucial.
The derivative of arcsin(x) is 1/√(1-x²).
The derivative of arccos(x) is -1/√(1-x²).
The derivative of arctan(x) is 1/(1+x²).
Chain Rule is used to differentiate composite function where inverse trigonometric function is introduced.
Use implicit differentiation when the equation can not be easily rearranged to y = f(x) format.

Bear in mind that integration is a process that reverses differentiation.
Integral of 1/√(a²-x²) results in arcsin(x/a).
Integral of -1/√(a²-x²) provides arccos(x/a).
Integral of 1/(a²+x²) becomes arctan(x/a).
Apply integration by substitution technique for some complex forms.
Utilise the method of integration by parts if the integral is a product of two functions.

Appreciate the fact that in practicality, physical phenomena like wave behaviour, pendulum motion and even medical imaging can be modelled using inverse trigonometric functions.
Through calculus involving inverse trigonometric functions, it is possible to gain insights and make accurate predictions about such models.
Understand the crucial role of arc length calculation, which accentuates the application of calculus on inverse trig functions.