INTEGRATION TECHNIQUES

INTEGRATION TECHNIQUES

1. Integration by parts

   • is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be derived in one line simply by integrating the product rule of differentiation.
     • The theorem can be derived as follows. Suppose u(x) and v(x) are two continuously differentiable functions. then you can integrate both sides with respect to x,then applying the definition of indefinite integral, and gives the formula for integration by parts.

2. Change of variable

   • In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule of differentiation.
     • The formula is used to transform one integral into another integral that is easier to compute. Thus, the formula can be used from left to right or from right to left in order to simplify a given integral. When used in the latter manner, it is sometimes known as u-substitution or w-substitution.

3. Partial Fractions

   • is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions.
     • There are two methods to solve the partial fractions
       • To decompose a fraction, you first factor the denominator. Let's work backwards from the example above. The denominator is x2 + x, which factors as x(x + 1).
         • Then you write the fractions with one of the factors for each of the denominators. Of course, you don't know what the numerators are yet, so you assign variables
           • Then you set the sum equal and Multiply things out, and group the x-terms and the constant terms:
             • For the two sides to be equal, the coefficients of the two polynomials must be equal. So you "equate the coefficients" to get aThis creates a system of equations that you can solve
       • Because there are equations where x has small values you just can skip to equal the equations.
         • Example
           • he equation "3x + 2 = A(x + 1) + B(x)" is supposed to be true for any value of x, we can pick useful values of x, plug-n-chug, and find the values for A and B. Looking at the equation "3x + 2 = A(x + 1) + B(x)", you can see that, if x = 0, then we quickly find that 2 = A: