![]() The antiderivative chain rule is used if the integral is of the form ∫u'(x) f(u(x)) dx. The chain rule of derivatives gives us the antiderivative chain rule which is also known as the u-substitution method of antidifferentiation. We know that antidifferentiation is the reverse process of differentiation, therefore the rules of derivatives lead to some antiderivative rules. Please do not confuse this power antiderivative rule ∫x n dx = x n+1/(n + 1) + C, where n ≠ -1 with the power rule of derivatives which is d(x n)/dx = nx n-1. Using the antiderivative power rule, we can conclude that for n = 0, we have ∫x 0 dx = ∫1 dx = ∫dx = x 0+1/(0+1) + C = x + C. Let us consider some of the examples of this antiderivative rule to understand this rule better. This rule is commonly known as the antiderivative power rule. Now, the antiderivative rule of power of x is given by ∫x n dx = x n+1/(n + 1) + C, where n ≠ -1. The antiderivative rules are common for types of functions such as trigonometric, exponential, logarithmic, and algebraic functions. We will discuss the rules for the antidifferentiation of algebraic functions with power, and various combinations of functions. If you are evaluating a definite integral with limits given to you on the integral sign, ALWAYS finish Step 6 by coming back to the expression written in terms of the original value (x or t or whatever) before substituting.In this section, we will explore the formulas for the different antiderivative rules discussed above in detail. Substitute the expression for u (or whatever) back into the integrated equation. Substitute, using your equations formed in Steps 1 and 3, for both components in the original equation. ![]() (for reasons I cannot explain - I use the letter which starts my name :-)).ĭifferentiate (separating the derivative fraction by placing the dx - or equivalent expression - on the right side). Identify the function with the highest power and let it equal a pronumeral of your choice - mostly we use the letter u The index differentiates to give the denominatorĪre integrated using the following steps: Step 1: The numerator is the derivative of the expression inside the brackets in the denominator. The numerator is the derivative of the expression in the denominator. The 2x is a derivative of the term inside the brackets. One part of their expression being the derivative of the other part. The integrals requiring the Reverse Chain Rule technique are identified by having The Reverse Chain Rule is the most common format for the functions we encounter. There are many formats for functions which we need to integrate. Then we replace the variable u with the original x term - i.e with f(x) - because the original question was expressed in terms of x. We now easily integrate with respect to the variable u. ![]() We then create another variable to represent the base part - so create f(x) = u = x 4 + 42. ![]() In the integral, (x 4 + 42) is the original or base compnent while (4x 3) is the derivative component. So if we have to integrate an expression of the form, we notice there are two parts: The technique combines these two parts by using another variable which makes the integration process more direct (and easier). The function should consist of two components with one component being the derivative of the other. The Reverse Chain Rule is a special technique for integrating a function having a particular structure. what the questions requiring the technique look like the concept of the Reverse Chain Rule technique Ģ. Where the techniques of Maths are explained in simple terms.Ĭalculus - Integration - Reverse Chain Rule.ġ.
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