Integration Methods to Know for AP Calculus AB/BC

Integration methods are essential for solving a variety of problems in AP Calculus AB/BC. These techniques, including basic rules, substitution, and integration by parts, help simplify complex integrals and deepen your understanding of calculus concepts.

1. Basic integration rules (power rule, constant multiple rule)

   • The power rule states that ∫x^n dx = (1/(n+1))x^(n+1) + C for n ≠ -1.
   • The constant multiple rule allows you to factor out constants: ∫k f(x) dx = k ∫f(x) dx.
   • Always remember to add the constant of integration (C) after integrating.

2. Integration by substitution (u-substitution)

   • Use u-substitution to simplify integrals by changing variables: let u = g(x), then dx = du/g'(x).
   • The goal is to rewrite the integral in terms of u, making it easier to evaluate.
   • After integrating, substitute back to the original variable to find the final answer.

3. Integration by parts

   • Based on the formula ∫u dv = uv - ∫v du, where u and dv are chosen parts of the integrand.
   • Choose u to be a function that simplifies when differentiated, and dv to be easily integrable.
   • This method is particularly useful for products of functions, such as polynomials and exponentials.

4. Trigonometric integrals

   • Integrals involving trigonometric functions often require specific identities to simplify.
   • Common techniques include using Pythagorean identities or converting to sine and cosine.
   • Look for patterns or symmetry in the integrand to facilitate integration.

5. Trigonometric substitution

   • Use trigonometric identities to simplify integrals involving square roots or quadratic expressions.
   • Common substitutions include x = a sin(θ), x = a tan(θ), or x = a sec(θ) based on the form of the integrand.
   • Remember to convert dx and the limits of integration if dealing with definite integrals.

6. Partial fraction decomposition

   • Break down rational functions into simpler fractions that can be integrated individually.
   • Ensure the degree of the numerator is less than the degree of the denominator before decomposition.
   • Use linear or quadratic factors in the denominator to set up the decomposition.

7. Integration of rational functions

   • Rational functions can often be integrated using partial fraction decomposition or polynomial long division.
   • If the degree of the numerator is greater than or equal to the denominator, perform long division first.
   • Simplify the resulting expression to facilitate integration.

8. Integration of inverse trigonometric functions

   • Familiarize yourself with the derivatives of inverse trigonometric functions, as they are often used in integration.
   • Common integrals include ∫(1/√(1-x^2)) dx = arcsin(x) + C and ∫(1/(1+x^2)) dx = arctan(x) + C.
   • Recognize when to apply these integrals based on the form of the integrand.

9. Integration using long division of polynomials

   • When integrating a rational function where the numerator's degree is higher than the denominator's, perform polynomial long division.
   • The result will be a polynomial plus a proper fraction, which can then be integrated separately.
   • This method simplifies the integral into manageable parts.

10. Improper integrals

    • Improper integrals occur when the limits of integration are infinite or the integrand approaches infinity within the interval.
    • Evaluate by taking limits: for example, ∫ from a to ∞ f(x) dx is evaluated as lim (t→∞) ∫ from a to t f(x) dx.
    • Determine convergence or divergence by analyzing the limit; if the limit exists, the integral converges.