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137 § 5.2 The Natural Log and Integration Theorem 5.5: Log Rule for Integration Let u be a differentiable function of x. 1 1. x dx ln | x | c 2. u dx ln | u | c u' Because the domain of ln does not include negative numbers we need to use the absolute value in the antiderivative. Example 3 i. Evaluate x dx ii. Evaluate 4 x 1 dx 1 138 iii. Evaluate using a change of variables. 2x ( x 1)2 dx iv. If a rational function has a numerator of degree greater than or equal to that of the denominator, division may reveal a form to which we can apply the Log Rule. x2 x 1 x2 1 dx §5.2: 9, 11, 18, 19 139 Guidelines for Integration 1. Memorize a basic list of the rules of integration. This includes the power rule, the log rule and the trigonometric rules. 2. Find an integration formula that resembles all or part of the integrand. Choose a satisfactory u by trial and error. 3. If you cannot find a satisfactory u BE CREATIVE! Try a trig identity, multiplication and division of the same quantity etc. 4. Try using technology to help solve the antiderivative symbolically. Example v. Evaluate vi. 1 x ln x dx tan xdx 140 Integrals of the 6 Basic Trigonometric Functions sin udu cos u c cos udu sin u c tan udu ln | cos u | c cot udu ln | sin u | c sec udu ln | sec u tan u | c csc udu ln | csc u cot u | c §5.2: 16, 22, 25, 27, 31, 34, 47 141 § 5.4 Exponential Functions 3 2 1 -4 -2 2 4 6 -1 -2 -3 Consider the function f ( x) ln x . Is increasing on its entire domain. Thus o f 1 ( x) ln( f 1 ( x)) x If x is a real number. But if x is a rational number, f 1 ( x) e x Definition: The inverse of the natural log function f ( x) ln x is called the natural exponentiation function and is denoted by f 1 ( x) e x That is y e x If and only if, x ln y 142 Example i. Solve 7 e x 1 ln 2 x 12 ii. Solve e Theorem 5.10: Operations with Exponential Functions Let a and b be real numbers. e e e a b a b ea e a b b e 143 Theorem 5.11: Derivative of e 1. d x e e x dx x 2. d u du e eu dx dx Example iii. Find d e 2 x 1 dx iv. Find d 3 x e dx §5.4: 1, 4, 8, 9, 11, 14, 21-24, 35, 37, 40, 41, 43, 45, 57, 61, 63 144 x Theorem 5.12: Integrating e Let u be a differentiable function of x. 1. e dx e x x c 2. Example v. Evaluate e vi. Evaluate 5xe 3 x 1 dx x2 dx e dx e u u c 145 vii. Evaluate viii. Evaluate 1 e 0 x dx x 3 e (4x )dx 4 146 e2 x dx ix. Evaluate 1 e2 x §5.4: 85, 87, 93, 95, 99 147 § 5.5 Bases Other Than e and Applications Definition of Exponential Function to Base a If ‘a’ is a positive real number (a 1) and x is any real number, then the exponential function to the base a is denoted a x and is defined by a x e(ln a ) x If a 1 , then y 1x 1. (Constant Function) Here are some familiar properties; a0 1 a x a y a x y ax a x y y a (a x ) y a xy Example i. The half life of carbon-14 is about 5730 years. If one gram of carbon-14 is present in a sample, how much will be in the sample in 10, 000 years? 148 Definition of Logarithmic function to Base a ‘a’ is a positive real number (a 1) and x is a positive real number, then the logarithmic function to base a is denoted log a x and is defined by log a x 1 ln x ln a Here are some familiar properties; log a 1 0 log a xy log a x log a y log a x log a x log a y y log a x n n log a x y a x if and only if x log a y a loga x x, x 0 log a a x x, for all x Example ii. Solve for x 5x 1 625 §5.5: 5, 9, 12, 13, 17, 21 log3 x 4 149 Theorem 5.13: Derivatives for Bases Other Than e Let ‘a’ be a positive real number (a 1) and let u be a differentiable function of x d a x (ln a )a x dx d du a x (ln a )a u dx dx d 1 log a x dx (ln a ) x d 1 du log a u dx (ln a )u dx Example iii. Find dy dx f ( x) 4 x §5.5: 38, 43, 46, 49, 53, 55 y log 10 2 x 150 Occasionally, an integrand involves an exponential function to a base other than e 1. Convert to base e a x e(ln a ) x 2. 1 a dx ln a a x x c Example iv. Evaluate 6 dx x Theorem 5.14: Power Rule for Real Exponents d n x nx n 1 dx n= real number d n du u nu n 1 dx dx u= differentiable function of x 151 Theorem 5.15: A Limit Involving e 1 x 1 lim 1 lim e x x x x x x Compound Interest Formulas P= amount of deposit t= number of years A= balance after t years r= annual interest rate (decimal form) n= number of yearly compounding periods Compounded n times per year r A P 1 n nt Compounded continuously A Pert §5.5: 61, 63, 65, 69, 71, 79, 85-88 152 § 5.7 Inverse Trig Functions and Differentiation Consider one of the six trig functions: sine, cosine, tangent, secant, cosecant, and cotangent All are periodic. Thus none are one-to-one. None of these six functions has an inverse! However, on certain defined intervals, the six trig functions can be one-to-one. On the interval , the sine function is increasing and thus one-to-one. 2 2 The inverse of the restricted sine function is y arcsin x if and only if sin y x Inverse Trig Functions Function y arcsin x iff sin y x Domain 1 x 1 Range y 2 2 y arccos x iff cos y x 1 x 1 0 y y arctan x iff tan y x x y arccot x iff cot y x x 0 y y arcsec x iff sec y x | x | 1 0 y , y y arc csc x iff csc y x | x | 1 2 2 y 2 2 y , y 0 2 iff= if and only if ** See the graphs of the six inverse trig functions on pg 371 153 Example i. Solve for x arctan(2 x 3) 4 Theorem 5.17: Derivatives of Inverse Trig Functions Let u be a differentiable function of x d u' arcsin u dx 1 u2 d u ' arccos u dx 1 u2 d u' arctan u 2 dx 1 u d u ' arc cot u 2 dx 1 u d u' arc sec u dx | u | u2 1 d u ' arc csc u dx | u | u2 1 154 Example ii. Find iii. Find d arcsin 2 x dx d arctan 3x dx 155 iv. Find d arc sec e 2 x dx **pg 376- Basic Differentiation Rules. This is a COMPLETE list of covered rules! § 5.7 1, 5, 7, 17, 19, 25-28, 43, 44, 48, 49, 53, 55