Integration+by+Substitution

Many integrals are most easily computed by means of a change of variables, commonly called a  //u//   **-substitution**. The substitution method applys the chain rule of derivatives but backwards. It undoes what the chain rule is. To integrate //f//(//g//(//x//))//g//'(//x//) //dx//
 * Integration by Substitution**
 * 1) Convert it to integral of //f//(//u//) //du// by substituting //u// for //g//(//x//) and substituting //du// for //g//'(//x//) //dx//, taking out a common numeric factor, //n//, if present.
 * 2) Evaluate (integral) //f//(//u//) //du// = //F//(//u//) + //c// or //n(integral)// //f//(//u//) //du// = //nF//(//u//) + //c//
 * 3) Replace //u// by //g//(//x//) in //F//(//u//)

There are occasions when it is possible to perform an apparently difficult piece of integration by first making a substitution. This has the effect of changing the variable and the integrand. When dealing with definite integrals, the limits of integration can also change. The substitution method turns an unfamiliar integral into one we can evaluate. In other words, substitution gives us a simpler integral involving the variable u. Let's now review the five steps for integration by substitution. **Step 1: ** Choose a new variable **u ** **Step 2: ** Determine the value **dx ** **Step 3: ** Make the substitution **Step 4: ** Integrate resulting integral **Step 5: ** Return to the initial variable **x **

Solution. It is clear that once we develop the through the binomial formula, we will get a polynomial function easy to integrate. But it is clear that this will take a lot of time with big possibility of doing mistakes !! Let us consider the substitution (the reason behind is the presence of //x// in the integral since the derivative of  is 2//x//). Indeed, we have //du// = 2//x dx// and therefor**e** We may check that the new integral is easier to handle since. Hence which does not complete the answer since the indefinite integral is a function of //x// not of //u//. Therefore, we have to go back and replace //u// by //u//(//x//):
 * Example Problem:**
 * Find**

Example problem 2 <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 18px; text-align: left; text-decoration: none; vertical-align: auto;">Integration by substituting <span class="blackC" style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 18px; text-decoration: none; vertical-align: auto;">__<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 18px; text-align: left; text-decoration: underline; vertical-align: auto;">u = ax + b __ <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">These are typical examples where we use the method of subsitution. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Example 1: <span class="normalF" style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: auto;">Evaluate <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Solution: <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Step 1: ** Chose a substitution function **<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">u ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">The substitution function is <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Step 2: ** Determine the value **<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">dx ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Step 3: ** Make the substitution <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Step 4: ** Integrate resulting integral <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Step 5: ** Return to the initial variable: **<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">x ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">So, the solution is:

**Work ﻿ Cited ** Khamsi, Muhamed A. "Integration by Parts: Example 2." //S.O.S Mathmatics//. Math Medics. Web. 24 May 2011.

[].

Petrović, Miloš. "Integration by Substitution." //MathPortal//. Web. 25 May 2011. <http://www.mathportal.org/calculus/integration-techniques/integration-by-

substitution.php>.


 * very good site to use for anything math related**
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