![]() When the antidifferentiation of the sum and difference of functions is to be determined, then we can do it by using the following formulas: Now, this rule is one of the easiest antiderivative rules. = 2x 2 + C Sum and Difference Antiderivative Rule An example using this antiderivative rule is: This implies, the antidifferentiation of kf(x) is equal to k times the antidifferentiation of f(x), where k is a scalar. To find the antiderivative of scalar multiple of a function f(x), we can find it using the formula given by, ∫kf(x) dx = k ∫f(x) dx. ∫du/v = u/v + ∫ dv Antiderivative Rule for Scalar Multiple of Function ![]() If f(x) = u and g(x) = v, then we have the antiderivatiev quotient rule as: Now, integrating both sides of the above equation, we haveį(x)/g(x) = ∫ dx Now, differentiating this we have,ĭ(f(x)/g(x))/dx = / 2 Another way to determine the antiderivative of the quotient of functions is, consider a function of the form f(x)/g(x). If the function includes algebraic functions, then we can use the integration by partial fractions method of antidifferentiation. The antiderivative quotient rule is used when the function is given in the form of numerator and denominator. = (x 2/2) ln x - x 2/4 + C Antiderivative Quotient Rule Then, according to the sequence above, the first function is ln x and the second function is x. This method is also commonly known as the ILATE or LIATE method of integration which is abbreviated of:įor example, we need to find the antiderivative of x ln x. The choice of the first function is done on the basis of the sequence given below. The formula for the antiderivative product rule is ∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) dx + C. It is one of the important antiderivative rules and is used when the antidifferentiation of the product of functions is to be determined. The antiderivative product rule is also commonly called the integration by parts method of integration. = sin (x 2) + C Antiderivative Product Rule Substitute this into the integral, we have Let us see an example and solve an integral using this antiderivative rule. 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. In this section, we will explore the formulas for the different antiderivative rules discussed above in detail.
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