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4.5 Integration by Substitution. "Millions saw the apple fall, but Newton asked why." -– Bernard Baruch . Objective. To integrate by using u-substitution. 2 ways…. Pattern recognition Change in variables Both use u-substitution… one mentally and one written out. Pattern recognition.

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## 4.5 Integration by Substitution

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**4.5 Integration by Substitution**"Millions saw the apple fall, but Newton asked why." -– Bernard Baruch**Objective**• To integrate by using u-substitution**2 ways…**• Pattern recognition • Change in variables • Both use u-substitution… one mentally and one written out**Pattern recognition**• Chain rule: • Antiderivative of chain rule:**Antidifferentiation of a Composite Function**• Let g be a function whose range is an interval I and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I then…**In other words….**• Integral of f(u)du where u is the inside function and du is the derivative of the inside function… If u = g(x), then du = g’(x)dx and**What about…**What is our constant multiple rule?**REMEMBER**• The contant multiple rule only applies to constants!!!!! • You CANNOT multiply and divide by a variable and then move the variable outside the integral sign. For instance…**In summary…**• Pattern recognition: • Look for inside and outside functions in integral • Determine what u and du would be • Take integral • Check by taking the derivative!**Guidelines for making a change of variables**• 1. Choose a u = g(x) • 2. Compute du • 3. Rewrite the integral in terms of u • 4. Evalute the integral in terms of u • 5. Replace u by g(x) • 6. Check your answer by differentiating**The General Power rule for Integration**• If g is a differentiable function of x, then • Equivalently, if u = g(x), then**Change of variables for definite integrals**• Thm: If the function u = g(x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g, then**Even and Odd functions**• Let f be integrable on the closed interval [-a,a] • If f is an even function, then • If f is an odd function then

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