Integrating with inverse trigonometric functions

Integration involving inverse trigonometric functions pertains to solving integrals that contain arcsin, arccos, arctan, arccot, arcsec, and arccsc.
A basic understanding of inverse trigonometric functions is necessary. These are mathematical functions that undo the effects of the corresponding trigonometric functions. For example, if sin(x) = y, then arcsin (y) = x.

Strong familiarity with differentiation rules of inverse trigonometric functions is vital as they frequently occur in the integration process. For example: the derivative of arcsin x is 1/(sqrt{1 - x^2}), for

<1, and the derivative of arctan x is 1/(1+x^2).

Often, the method of integration by parts (the integral analog of the product rule in differentiation) is implemented. This strategy relies on choosing appropriate ‘parts’ for the integration, generally selecting a function that simplifies on differentiation as part of du (in the formula ∫udv = uv - ∫vdu).
Occasionally, the method of substitution might be employed. The main idea is to change the variable in the integral to simplify the problem, often using a trigonometric identity to do so.
Regular practice leads to becoming more adept at recognising the instances where inverse trigonometric functions can be used as part of the integration process to simplify the problem.
Mastery of useful trigonometric identities, such as sin^2x + cos^2x = 1, is hugely beneficial as they often contribute to simplifying otherwise complex integrals.
Remember, the properties of definite integrals often come into play. For example, the property ∫ from a to b [f(x)]dx = - ∫ from b to a [f(x)]dx is used often in these problems.
Final note: always keep in mind that inverse trigonometric functions can yield results in different quadrants, depending on the value of the argument. Understanding the quadrant-specific behaviour of these functions will help in arriving at the correct answer.