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ACOW INDEFINITE INTEGRALS MODULE Updated 5/28/2016 Page 1 of 2 Activity A2- Integration The process of antidifferentiation is also known as integration. Integration uses the symbol and is known as the integral sign. When this integral sign is used, we always find the most general antiderivative of the function that comes after the integral sign. For example x 2 dx means find the most general antiderivative of x 2 . So, x dx 3 x 2 1 3 C . The function x 2 is called the integrand and the arbitrary constant C is called the constant of integration. (Note: The dx that follows the x 2 in the example above tells you to use the power rule for the variable x. Mathematically we say “Integrate x 2 with respect to x”.) 1. Find t 3 dt So far, we have only found the antiderivative of functions that are of the form x n . To find the antiderivatives of more complicated functions we need some additional integration formulas(RS2). 2. State the correct rule (see list below) that justifies each step. Choose the letter that corresponds to the rule. a) Integration of a constant b) Integration of a constant times a function c) Integration of the sum of functions d) Integration of the difference of functions e) Power Rule for Antiderivatives 2 1/ 2 2 1/ 2 4 x 5 x 2 dx 4 x dx 5x dx 2 dx (answer blank goes here) 4 x 2 dx 5 x1/ 2 dx 2 dx (answer blank goes here) 4 x 2 dx 5 x1/ 2 dx 2 x C (answer blank goes here) Find the following. 3. 3 x 2 7 x 8 dx 4. 1 3 5 x2 4x2 5 2 dx 3 x 4 3 10 3/ 2 x x 2x C 3 3 (answer blank goes here) ACOW INDEFINITE INTEGRALS MODULE 5. 4 x Updated 5/28/2016 Page 2 of 2 6 x 3 dx 3 There are two other special rules for integration(RS3) that involve the natural logarithm and the exponential function. Recall that the general power rule for antidifferentiation stated n 1 . The special rule for integration now allows us to integrate functions with an exponent of –1. Find the following. 1 2 6. x 2 e x 5 dx x 3 7. 3e x 4 1 3 2 x dx x x3 Sometimes we can determine the value of the constant of integration if we are given some information about the original function. For example if we know that the 1 antiderivative of f ( x) is F ( x) x 2 x C and we are given F (4) 3 , then we can 2 find the value of C by setting up and solving the following equation: 1 3 (4) 2 4 C 2 1 3 (16) 4 C 2 3 8 4 C 3 4 C 7 C 1 Thus, the antiderivative of f ( x) is F ( x) x 2 x 7 2 In the problems that follow, let F '( x) f ( x) . 8. Find the antiderivative of f ( x) 3x 2 6 x 5 given F (1) 7 . 2 1 9. Find the antiderivative of f ( x) x 4 2 e x given F (0) 2 . 3 x 10. Find the antiderivative of f ( x) ln( x) 3x 2 e x given F (1) e .