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This guide explores the concept of "u" substitution in definite integrals and its application through the First Fundamental Theorem of Calculus. We'll break it down into two steps: first, finding the function F(x), and then determining its derivative F'(x). The theorem states that if f is continuous on the interval [a,b], then it is differentiable at every point in that interval. We also examine the effects of substitution on integration limits and provide several examples to clarify these concepts.
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Definite Integrals 5.6 “u” substitution
The First Fundamental Theorem of Calculus To understand this, lets do a problem in 2 steps: • Find F(x): • Find F’(x)
The First Fundamental Theorem of Calculus If f is continuous on [a,b] then is differentiable on every point in [a,b] and
u substitution We’ve talked about this a little; as you substitute “u” in, the limits of integration will change. Lets look at a few examples to see how these work:
Do you have to do this? No, you don’t. It just sometimes makes some very unwieldy problems look nicer and simplify quicker.
