# Calculo integral

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• Publicado : 17 de mayo de 2011

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Contents
1 Functions 1.1 The Concept of a Function . . . . . . . . . . . . . . 1.2 Trigonometric Functions . . . . . . . . . . . . . . . 1.3 Inverse Trigonometric Functions . . . . . . . . . . . 1.4 Logarithmic, Exponential and Hyperbolic Functions 2 Limits and Continuity 2.1 Intuitive treatment and deﬁnitions . . 2.1.1 Introductory Examples . . . . . 2.1.2 Limit: Formal Deﬁnitions . . . 2.1.3Continuity: Formal Deﬁnitions 2.1.4 Continuity Examples . . . . . . 2.2 Linear Function Approximations . . . . 2.3 Limits and Sequences . . . . . . . . . . 2.4 Properties of Continuous Functions . . 2.5 Limits and Inﬁnity . . . . . . . . . . . 3 Diﬀerentiation 3.1 The Derivative . . . . . . . . . . . 3.2 The Chain Rule . . . . . . . . . . . 3.3 Diﬀerentiation of Inverse Functions 3.4 ImplicitDiﬀerentiation . . . . . . . 3.5 Higher Order Derivatives . . . . . . 2 2 12 19 26 35 35 35 41 43 48 61 72 84 94 99 99 111 118 130 137

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4 Applications of Diﬀerentiation 146 4.1 Mathematical Applications . . . . . . . . . . . . . . . . . . . . 146 4.2 Antidiﬀerentiation .. . . . . . . . . . . . . . . . . . . . . . . 157 4.3 Linear First Order Diﬀerential Equations . . . . . . . . . . . . 164 i

ii 4.4 4.5 5 The 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

CONTENTS Linear Second Order Homogeneous Diﬀerential Equations . . . 169 Linear Non-Homogeneous Second Order Diﬀerential Equations 179 Deﬁnite Integral Area Approximation . . . . . . . . . . . The Deﬁnite Integral . .. . . . . . . . . Integration by Substitution . . . . . . . . Integration by Parts . . . . . . . . . . . Logarithmic, Exponential and Hyperbolic The Riemann Integral . . . . . . . . . . Volumes of Revolution . . . . . . . . . . Arc Length and Surface Area . . . . . . 183 183 192 210 216 230 242 250 260

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6 Techniques of Integration 6.1 Integration by formulae . . . . . 6.2 Integration by Substitution . . . 6.3 Integration by Parts . . . . . . 6.4 Trigonometric Integrals . . . . . 6.5 Trigonometric Substitutions . . 6.6 Integration by Partial Fractions 6.7 Fractional PowerSubstitutions . 6.8 Tangent x/2 Substitution . . . 6.9 Numerical Integration . . . . .

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267 . 267 . 273 . 276 . 280 . 282 . 288 . 289 . 290 . 291 294 294 299 304 314

7 Improper Integrals and Indeterminate Forms 7.1 Integrals over Unbounded Intervals . . . . . . 7.2 Discontinuities at End Points . . . . . . . . . 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Improper Integrals . . . . . . . . . . . . . . . 8 Inﬁnite Series 8.1 Sequences . . . . . . . .. . . 8.2 Monotone Sequences . . . . . 8.3 Inﬁnite Series . . . . . . . . . 8.4 Series with Positive Terms . . 8.5 Alternating Series . . . . . . . 8.6 Power Series . . . . . . . . . . 8.7 Taylor Polynomials and Series

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