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**Unformatted text preview: **Lecture 32 1 Lecture 32: (Sec. 6.2 and 6.3) Integration by Substitution, Part II Recall the following results: Let f be a function of u , where u is a differentiable function of x . Then if F is an antiderivative of f ( u ), integraldisplay f ( u ) du dx dx = Integration by Substitution with exponential and log- arithmic functions. We have two more rules of substitution: I. Integrals of Exponential Functions ex. integraldisplay e 3- 4 x dx = Lecture 32 2 ex. integraldisplay e 2 x (1- e 2 x ) 2 dx Lecture 32 3 II. Integrals involving Logarithmic Func- tions integraldisplay 1 x dx = integraldisplay 1 u du dx dx = ex. integraldisplay 2 x 2- 4 x 3- 6 x + 3 dx = Lecture 32 4 Integrals involving ln x ex. integraldisplay ln x x dx = ex. integraldisplay 1 x ln x dx = Lecture 32 5 (Sec. 6.3, Part I) Area and the Definite Integral; The Fun- damental Theorem of Calculus How to find the area under the graph of f on the interval [ a, b ] or from a to b ?...

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