Study Guide “U” Substitution Calculus By: Eldon S. (edited very slightly by Mr W) Prerequisites: In order to correctly and effectively use “u” substitution, one must know how to do “basic” integration and derivatives as well as know the basic patterns of derivatives and integrals (for example, the derivative of sin (x) is cos (x) dx). Purpose: The most important aspect of “u” substitution to remember is that “U” substitution is meant to make the integral EASIER to solve. If it gets harder, it has been done wrong. (The best way to explain “U” Substitution is through examples so the rest of this study guide will comprehensively go through an example problem and relay every step along the way.) Side notes: a[int]b is the same as the integral from a to b. [int] is just integral without number limits. Example 1 Solve the problem ∫6x3(3x4+1)5dx Step 1: Identifying the “u” The first step in “u” substitution is identifying the part of the function that will be represented by u. This first step is often the hardest because it takes a little instinct and prediction. Basically, the variable “u” is meant to replace the part of the function that is the “original” part of the function. In the integral function, one part is the “original” and the other part is the derivative part (for example, in ∫f(x)dx, f(x) is the original and dx is the derivative). In example 1, the u will replace the part of the function in the parenthesis because that part is the “original” part and the part of the function on the outside of the parenthesis “looks like” the derivative of the inside (the derivative vs. original idea will make more sense in steps 2 and 3). ∫ 6x3(3x4+1)5dx = ∫6x3(u)5dx u = 3x4+1 Step 2: Solving the derivative of u or “du” The second step of “u” substitution is taking the derivative of u u = 3x4+1 du = 12x3dx Step 3: Using u and du The third step is figuring out how to substitute u and du back into the integral. Substituting u is pretty straightforward but du can take some rearranging. The derivative of u has to be made so that it matches the part of the original integral that isn’t u. In example 1, du needs to be made to match the part of the problem not part of u or (6x3dx). {du = 12x3dx} = {du/2 = 6x3dx} Step 4: Substitution and solve The fourth step is now substituting u and du in order to solve the equation. ∫6x3(3x4+1)5dx = ∫ (u)5du/2 = (1/2) ∫ (u)5du Now that the integral has become simplified, it is time to take the integral. (1/2) ∫ (u)5du = (1/2) * (1/6)(u)6 + C = (1/12)(u)6 + C The last part is to substitute the value of u back into the solution. (1/12)(u)6du = (1/12)(3x4+1)6 + C = The answer Conclusion: In summation, “u” substitution is a method that is used to solve complex integrals through creating simple “u” integral problems and then substituting the original values back in.