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Appendix D Theory

1.1 Antiderivatives

Theorem 1.1.7 General antiderivative formula

Theorem 1.1.14 Antiderivative formulas

Theorem 1.1.15 Antiderivative rules

1.3 Riemann sums

Theorem 1.3.4 Summation formulas

Theorem 1.3.5 Summation rules

1.4 Definite integral

Theorem 1.4.5 Integrable functions

Theorem 1.4.6 Integral properties

1.5 Fundamental theorem of calculus

Theorem 1.5.1 Fundamental theorem of calculus

Theorem 1.5.8 FTC II: rate of change version

1.6 Fundamental theorem (cont.)

Corollary 1.6.4 Continuous functions have antiderivatives

Theorem 1.6.5 FTC I: geometric

1.7 Substitution

Theorem 1.7.1 Substitution

1.10 Inverse functions

Theorem 1.10.3 Horizontal line test

Theorem 1.10.5 Monotonic functions are one-to-one

Theorem 1.10.7 Inverse function compendium

Theorem 1.10.8 Derivative formula for inverses

1.11 The natural logarithm

Theorem 1.11.2 Natural logarithm properties

Corollary 1.11.4 Antiderivative of \(f(x)=1/x\)

Theorem 1.11.6 Trigonometric antiderivatives

1.12 Exponential functions

Theorem 1.12.2 Exponential function properties

Theorem 1.12.5 Logarithmic and exponential compendium

Theorem 1.12.7 Logarithmic change of base

Theorem 1.12.8 Derivative/antiderivative compendium

1.14 L’Hôpital’s rule

Theorem 1.14.4 L’Hôpital’s rule

1.16 Inverse trigonometric functions

Theorem 1.16.3 Properties of inverse trigonometric functions

Theorem 1.16.7 Derivatives of inverse trig functions

1.17 Integration strategies

Theorem 1.17.1 Idefinite integral compendium

1.18 Integration by parts

Theorem 1.18.1 Integration by parts

1.19 Trigonometric integrals

Theorem 1.19.1 Trigonometric identities

1.21 Rational functions

Theorem 1.21.2 Polynomial facts

Theorem 1.21.3 Partial fraction decomposition

1.23 Numerical integration: error bounds

Theorem 1.23.1 Error bounds

1.24 Improper integrals

Theorem 1.24.12 Improper integrals: \(p\)-test

1.25 Improper integrals: convergence tests

Theorem 1.25.1 Direct comparison test

Theorem 1.25.4 Limit comparison test