INTEGRATION TECHNIQUES

INTEGRATION TECHNIQUES

1. INTEGRATION BY PARTS

   • Integration by parts is usually used in exponential and logarithmic functions.It is of high importance to find two important parts given in the integral, these being the function u and the differential dv. The remaining elements will be dv .
     • Now that we have identified our values we need to seek the values of du and v consecutively.
       • After we have done that we arrange the values with the formula seen below.
         • • In the end after we have substituted everything we only need to solve and simplify to obtain our answer.
     • To choose the one that will be u we use LATE (Logarithmic/ Algebraic/Trigonometric/Exponential) .
   • In certain cases
     • a single use of integration by parts will not be enough, to solve it we will have to run the process several times.
       • We would use the tabulation method, in which we make a little graph that serves as a shortcut in which we place the following, signs that start with + and alternate/ Function u ands its derivative/ and the differential dv and it’s antiderivatives.
         • We calculate the derivative until we obtain 0 as a derivative.

2. DOUBLE SUBSTITUTION

   • If we have two terms that are being multiplied and we want to integrate them, but by using the generalized p. rule we can’t cancel anything we have to use double substitution to solve it.
     • First we need to define u for our change of variables.
       • Then we need to differentiate u to find du and solve for dx.
         • hen we need to find x so all of our expression is in terms of u. After that we substitute and simplify.
           • In the end what we have to do is integrate and substitute back for u so in our final answer we have everything in terms of x.

3. PARTISL FRACTIONS

   • If the integral you are trying to solve is within a fraction and if you integrate it you wouldn't be able to cancel anything, you need to use partial fractions.
     • If we want to solve the integral by partial fractions we need to factor the denominator first.
       • Then we need to rewrite the expression to cancel the denominator and incorporate our A,B,C variables ( the amount of these may be different from problem to problem).
         • Then we need to find the values for these new variables we just assigned.
           • In the end we substitute the values we got for the variables and solve by using the logarithmic rule.

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