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Section 6.1: Antiderivatives and slope fields

Section 6.1: Antiderivatives and slope fields. The Indefinite Integral. Find: ∫4t 3 +6t 2 -5 dt. ∫ax n dx=(ax n+1 ) /(n+1) Use the power rule in reverse. t 4 +2t 3 -5t Here is your antiderivative.

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Section 6.1: Antiderivatives and slope fields

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  1. Section 6.1: Antiderivatives and slope fields

  2. The Indefinite Integral Find: ∫4t3+6t2-5 dt ∫axn dx=(axn+1)/(n+1) Use the power rule in reverse. t4+2t3-5t Here is your antiderivative. There is one more step. Like a derivative represents a slope of a given function, an antiderivative represents the function with a given slope. Because the slope of a function tells us nothing about its location, an antiderivative can actually represent infinitely many functions that are essentially the same curve shifted up or down. This is represented by adding an integration constant “+C” to the end of an indefinite integral. ∫4t3+6t2-5dt = t4+2t3-5t+C

  3. Integrating With an Initial Condition Find: ∫4t3+6t2-5dt Given: f(1) = 0 ∫4t3+6t2-5dt = t4+2t3-5t+C Begin with the indefinite integral (taken from previous example) Because we are given a value of the function at a certain value of t, we can determine the value of C by plugging the numbers into the equation. 0 = 14+2(1)3-5(1)+C Substitute 1 for t and set equal to 0. C=2 Solve for C. • f(t) = t4+2t3-5t+2 Plug C into the original equation.

  4. Applying some Properties of Integrals Find: ∫2f(x) - 3g(x) dx Given: ∫f(x) = 4x and ∫g(x) = 3x • ∫2f(x) dx -∫3g(x)dx Split into two integrals. • 2∫f(x) dx -3∫g(x)dx Pull out the constants. • 2(4x)- 3(3x) Plug in the given values. ∫2f(x) - 3g(x) dx = -x Simplify.

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