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积分结果为反三角函数的积分
有两个分式的积分结果是反三角函数：1 除以根号下 a 的平方减 x 的平方，积分结果为反正弦函数；1 除以 a 的平方加 x 的平方，积分结果为反正切函数。至此，初等积分表得以补全。
1. The two gateway integrals
The two gateway integrals
2. Pattern recognition and scaling
Pattern recognition and scaling
3. Completing the square
Completing the square
4. Where these integrals arise
Where these integrals arise
5. Arcsine pattern shape
Which integrand has the arcsine integral form?
6. Arctangent scaling
What is the indefinite integral of dx/(4 + x^2) with respect to x?
7. Completing the square purpose
In the integral of dx/(x^2 + 6x + 13), what does completing the square accomplish?
8. Which pattern for a sum under root
The integrand 1/sqrt(a^2 + x^2) does not give an arcsine. To which family does its integral belong?
9. Arcsine value
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.
10. Sum versus difference discrimination
Which single feature of the denominator decides between the arcsine and the arctangent gateway formulas?
11. Arctangent integral at a equals one## practice-fill Arctangent integral at a equals one
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.
12. Completed square form
Completing the square in x^2 + 6x + 13 produces (x + 3)^2 plus what constant? Answer with a number.
13. Linear inner factor
The integral of dx/sqrt(9 - 4x^2) is evaluated by substitution u = 2x. What is the result?
14. Gateway formula count## practice-fill Gateway formula count
This section's table completes the elementary integrals with the arcsine and arctangent forms. How many gateway formulas are there? Answer with a number.