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Honors Finite – Sec. 11.1- Antiderivatives and Indefinite Integrals Of x 2 5? What is the derivative of x 2 ? Of x 2 8? Notice they all have the same derivative. All of the above expressions are antiderivatives of 2x. The most general antiderivative of 2x is x 2 C , where C is any constant. The most general antiderivative is called the indefinite integral . 2xdx x 2 C An indefinite integral can be checked by differentiation. f ( x)dx g ( x) C if and only if g '( x) f ( x). Power Rule for Integration 1 x dx n 1 x n n 1 C , n 1 , To integrate x to a power, add 1 to the power and divide by the new power. Example 1: x dx 4 Example 2: What happens when you try this problem? 1 3 x dx x 1 Example 3: 1dt dx Other Rules: 1. The integral of a sum is the sum of the integrals. 2. The integral of a constant times a function is the constant times the integral of the function. 3. The integral of a constant is the constant times x + C. 3 Example 4: (14 Example 5: x 2 Example 6: 4 x4 4 x2 x dx x Example 7: x2 1 x 1 dx 4 x3 2 4 x3 )dx dx 4 3 1 2 Example 8: A company’s marginal cost function is MC 21x 6 x 50 , where x is the number of units, and fixed costs are $3000. Find the cost function.