INTEGRATION METHODS

INTEGRATION METHODS

1. Variable Change Integration

   • It provides us with a process that allows us to recognize when an integrand is the result of a derivative in which the chain rule has been used. Let f (x) be the function we wish to integrate, then we make the following variable change: x = g (t), d (x) = g '(t) dt, with which:
     • • For the variable change formula to have a chance of success, we must identify in the integrating a function u and a u '(its derivative).

2. Integration by parts

   • This method will allow us to solve integrals of functions that can be expressed as a product of one function by the derivative of another. Let u and v be two continuous, derivable functions and their derivatives du and dv are integrable, then: u = f (x), v = g (x), then du = f '(x) dx, dv = g' (x) dx

3. Integration of rational functions

   • If the degree of P (x) is greater than or equal to the degree of Q (x).
   • If the degree of P (x) is less than the degree of Q (x).

4. Trigonometric Integration Techniques

   • 1. Powers of sines and cosines.
     • • To solve this type of integrals, we consider two cases: If n is odd, that is, n = 2k + 1, we factor the integrand, for example: sennx dx = sen2k + 1x dx = (sen2x) k senx dx We use the identity sen2x + cos2x = 1 and take the following variable change: - In case of breast powers: u = cosx - In case of cosine powers: u = senx
   • 2. Products of sines and cosine powers.
     • • If m and n are even, we will use the identities: sen2x = (1-cos2x) / 2 and cos2x = (1 + cos2x) / 2 If m or n is odd, we will use the identity: sen2x + cos2x = 1
   • 3. Trigonometric substitution.
     • 1) If a radical of the form appears in the integrand:
       • • we take the change of variable: x = a sin θ, with a> 0; θ = arcsenx
     • 2) If a radical of the form appears in the integrand:
       • • we take the following variable change: x = a tan θ, with a> 0;θ = arctanx
     • 3) If a radical of the form appears in the integrand:
       • • we take the following variable change: x = a sec θ, with a> 0 θ = arcsec (x / a)

5. ·

   • The area of ​​the portion of the plane that is limited by the function, the horizontal axis OX and the vertical lines of equations x = a and x = b is called a definite integral of the function between points a and b.
     • Definite integral