Some integrals do not simplify into powers, exponentials, or ordinary trig functions, but they match patterns whose antiderivatives are inverse trig functions. These results matter because they appear in motion, geometry, electric fields, probability, and many substitution problems. The main skill is recognizing the algebraic shape of the denominator or radical before trying a long method.
Once the shape is identified, the integral often becomes a direct formula with a small substitution.
Key Facts
- ∫ dx / sqrt(a^2 - x^2) = arcsin(x/a) + C, for a > 0
- ∫ dx / (a^2 + x^2) = (1/a) arctan(x/a) + C, for a > 0
- ∫ dx / (x sqrt(x^2 - a^2)) = (1/a) arcsec(|x|/a) + C, for a > 0
- If u = g(x), then ∫ g'(x) dx / sqrt(a^2 - g(x)^2) = arcsin(g(x)/a) + C
- Complete the square to rewrite quadratics, such as x^2 + 6x + 13 = (x + 3)^2 + 4
- Always include the constant of integration: antiderivatives have the form F(x) + C
Vocabulary
- Antiderivative
- An antiderivative of f(x) is a function F(x) whose derivative is f(x).
- Inverse trigonometric function
- An inverse trigonometric function, such as arcsin, arctan, or arcsec, gives an angle from a trigonometric ratio.
- Standard form
- A standard form is a recognizable algebraic pattern that matches a known integration formula.
- Completing the square
- Completing the square rewrites a quadratic as a squared binomial plus or minus a constant.
- Substitution
- Substitution replaces part of an integral with a new variable to make the integral match a simpler formula.
Common Mistakes to Avoid
- Using arcsin for a^2 + x^2. This is wrong because the plus form in the denominator matches arctan, while arcsin comes from sqrt(a^2 - x^2).
- Forgetting the factor from substitution. If u = 3x, then du = 3 dx, so the integral needs a factor of 1/3 to stay equivalent.
- Not completing the square before choosing a formula. A quadratic such as x^2 + 4x + 8 must be rewritten as (x + 2)^2 + 4 before the arctan form is visible.
- Dropping the absolute value in the arcsec form. The standard result uses arcsec(|x|/a) because the derivative depends on the domain and sign of x.
Practice Questions
- 1 Evaluate ∫ dx / sqrt(25 - x^2).
- 2 Evaluate ∫ dx / (x^2 + 6x + 13).
- 3 Explain how you would decide whether an integral should lead to arcsin, arctan, or arcsec when the integrand contains a square root or quadratic expression.