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4-5 INTEGRATION BY SUBSTITUTION MS. BATTAGLIA – AP CALCULUS THM 4.13 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 ò f (g(x))g'(x)dx = F(g(x)) + C Letting u = g(x) gives du = g’(x)dx and ò f (u)du = F(u) + C EXPLORATION The integrand in each of the following integrals fits the pattern f(g(x))g’(x). Identify the pattern and use the result to evaluate the integral. ò 2x(x 2 +1) dx 4 ò 3x 2 x +1dx 3 The next three integrals are similar to the first three. Show how you can multiply and divide by a constant to evaluate these integrals. x x +1 dx ( ) ò 2 4 òx 2 x +1dx 3 RECOGNIZING THE F(G(X))G’(X) PATTERN Find 5 cos 5x dx ò MULTIPLYING AND DIVIDING BY A CONSTANT Find x x +1 dx ( ) ò 2 2 CHANGE IN VARIABLES If u=g(x) then du=g’(x)dx and the integral takes the form ò f (g(x))g'(x)dx = ò f (u)du = F(u) + C CHANGE OF VARIABLES ò 2x -1 dx CHANGE OF VARIABLES Find x 2x -1 dx ò CHANGE OF VARIABLES sin 3x cos 3x dx ò 2 GUIDELINES FOR MAKING A CHANGE OF VARIABLES 1. Choose a substitution u=g(x). Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power. 2. Compute du=g’(x)dx 3. Rewrite the integral in terms of the variable u. 4. Finding the resulting integral in terms of u. 5. Replace u by g(x) to obtain an antiderivative in terms of x. 6. Check your answer by differentiating. THE GENERAL POWER RULE FOR INTEGRATION If g is a differentiable function of x, then ò g(x)] [ [ g(x)] g'(x)dx = n+1 n n +1 + c, n ¹ -1 Equivalently, if u=g(x), then n+1 u u du = + c, n ¹ -1 ò n +1 n SUBSTITUTION AND THE GENERAL POWER RULE 3(3x -1) dx ò 4 (2x +1)(x + x)dx ò 2 -4x dx 2 ò (1- 2x ) SUBSTITUTION AND THE GENERAL POWER RULE 3x ò x - 2 dx 2 3 cos xsin x dx ò 2 CHANGE OF VARIABLES FOR DEFINITE INTEGRALS 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 ò b a f (g(x))g'(x)dx = ò g(b) g(a) f (u)du CHANGE OF VARIABLES ò 1 0 x ( x +1) dx 2 3 CHANGE OF VARIABLES ò 5 1 x dx 2x - 1 INTEGRATION OF EVEN AND ODD FUNCTIONS Let f be integrable on the closed interval [-a,a] 1. If f is an even function, then 2. If f is an odd function, then ò a -a ò a -a a f (x)dx = 2 ò f (x)dx 0 f (x)dx = 0 INTEGRATION OF AN ODD FUNCTION sin ( ò p /2 - p /2 3 x cos x + sin x cos x )dx HOMEWORK Read 4.5 Page 306 #2, 3, 7-10, 13, 29, 31, 33, 37, 39, 47, 53, 107