Integrals Resulting in Inverse Trigonometric Functions

The two gateway integrals

Pattern recognition and scaling

Completing the square

Where these integrals arise

Which integrand has the arcsine integral form?

What is the indefinite integral of dx/(4 + x^2) with respect to x?

In the integral of dx/(x^2 + 6x + 13), what does completing the square accomplish?

The integrand 1/sqrt(a^2 + x^2) does not give an arcsine. To which family does its integral belong?

The integral of dx/sqrt(1 - x^2) from 0 to 1/2 equals arcsin(1/2). What number is this? Answer with a fraction like pi/6.

Which single feature of the denominator decides between the arcsine and the arctangent gateway formulas?

The integral of dx/(1 + x^2) from 0 to 1 equals arctan(1). What number is this, as a fraction of pi? Answer like pi/4.

Completing the square in x^2 + 6x + 13 produces (x + 3)^2 plus what constant? Answer with a number.

The integral of dx/sqrt(9 - 4x^2) is evaluated by substitution u = 2x. What is the result?

This section's table completes the elementary integrals with the arcsine and arctangent forms. How many gateway formulas are there? Answer with a number.