Taking sin(2x) as an example, I would say something along the lines of 'integrating the outside function of sin(2x) gives -cos(2x) but since this gives 2sin(2x) on differentiation the final answer needs to be -1/2 cos(2x)'. My teacher has been keen to make us better at integrating quickly by inspection without having to go through u substitution each time for easy integrals so the general strategy we have learned for these problems is to pretend you are applying the chain rule in reverse - that is integrate the outside function with respect to the inside function and then adjust for the chain rule by dividing by the derivative of the inner function. (3x+5) 2 or sin(2x) ), it is often best to think about what you need to differentiate to get this original function. I have been taught that when you have a simple integral such as only having one inner function (e.g. Please e-mail any correspondence to Duane Koubaīy clicking on the following address About this document. Your comments and suggestions are welcome.
For example, if u = x+1, then x= u-1 is what I refer to as a "back substitution".Ĭlick HERE to see a detailed solution to problem 13.Ĭlick HERE to see a detailed solution to problem 14.Ĭlick HERE to see a detailed solution to problem 15.Ĭlick HERE to see a detailed solution to problem 16.Ĭlick HERE to see a detailed solution to problem 17.Ĭlick HERE to see a detailed solution to problem 18.Ĭlick HERE to return to the original list of various types of calculus problems. I call this variation a "back substitution". The following problems require u-substitution with a variation. Make careful and precise use of the differential notation dx and du and be careful when arithmetically and algebraically simplifying expressions.Ĭlick HERE to see a detailed solution to problem 1.Ĭlick HERE to see a detailed solution to problem 2.Ĭlick HERE to see a detailed solution to problem 3.Ĭlick HERE to see a detailed solution to problem 4.Ĭlick HERE to see a detailed solution to problem 5.Ĭlick HERE to see a detailed solution to problem 6.Ĭlick HERE to see a detailed solution to problem 7.Ĭlick HERE to see a detailed solution to problem 8.Ĭlick HERE to see a detailed solution to problem 9.Ĭlick HERE to see a detailed solution to problem 10.Ĭlick HERE to see a detailed solution to problem 11.Ĭlick HERE to see a detailed solution to problem 12. Most of the following problems are average. Here is another illustraion of u-substitution. In essence, the method of u-substitution is a way to recognize the antiderivative of a chain rule derivative. Of course, it is the same answer that we got before, using the chain rule "backwards". Make substitutions into the original problem, removing all forms of x, resulting in Is an arithmetic fraction, and multiply both sides of the previous equation by dx getting Now "pretend" that the differentiation notation Now the method of u-substitution will be illustrated on this same example. This is an illustration of the chain rule "backwards".
#Chain rule integration how to#
However, it may not be obvious to some how to integrateĬan be computed using the chain rule and is For example, since the derivative of e x is This method is intimately related to the chain rule for differentiation. The method of u-substitution is a method for algebraically simplifying the form of a function so that its antiderivative can be easily recognized. We will assume knowledge of the following well-known, basic indefinite integral formulas : It is a method for finding antiderivatives. The following problems involve the method of u-substitution.
0 kommentar(er)
