Calculus: Learning Worksheets Chapter 5 Name ________________________________ Date ______________ Class ____________ Section 5-1 Antiderivatives and Indefinite Integrals Goal: To find antiderivatives and indefinite integrals of functions using the formulas and properties Theorem 1 Antiderivatives If the derivative’s of two functions are equal on an open interval (a, b), then the functions differ by at most a constant. Symbolically, if F and G are differentiable functions on the interval (a, b) and F ‘( x) G ‘( x) for all x in (a, b), then F ( x) G ( x) k for some constant k. Formulas and Properties of Indefinite Integrals For C and k both a constant 1. 2. 3. 4. 5. x n1 C, n 1 x dx n 1 x x e dx e C 1 x0 dx ln x C , x kf ( x) dx k f ( x) dx [ f ( x) g ( x)]dx f ( x)dx g ( x)dx n In Problems 1–3, find each indefinite integral and check by differentiating. 1. ∫ 26x dx 161 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 1 2. 9x 2 dx 3. ∫ 7e x dx 4. Find all the antiderivatives for dy = 7 z −1 + 3. dz 162 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 In Problems 5–8, find each indefinite integral. 5. ∫ x5 ( x3 + 7 x 2 + 6) dx 3 6. 2 x5 dx x ⎛ 9x + 2 ⎞ 7. ∫ ⎜ 4 ⎟ dx ⎝ x ⎠ 163 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 5 x5 3 x 6 e x 8. dx x6 In Problems 9–12, find the particular antiderivative of each derivative that satisfies the given conditions. 9. R ‘( x) = 4 x3 + 12 x 2 − 5; R (2) = 50 164 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 10. dy 5et 4t 8; dt Chapter 5 y (0) 8 165 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 11. dD 6 x3 + 5 x = ; dx x3 Chapter 5 D(10) = 50 166 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 12. h ‘( x) 6 x 1 7 x 2 ; h(1) 3 167 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 168 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 Name ________________________________ Date ______________ Class ____________ Section 5-2 Integration by Substitution Goal: To find the indefinite integrals using general indefinite integral formulas Formulas: General Indefinite Integral Formulas [ f ( x)]n1 C, 1. [ f ( x)] f ‘( x)dx n 1 2. e f ( x ) f ‘( x)dx e f ( x ) C 1 f ‘( x)dx ln f ( x) C 3. f ( x) n u n1 C, 4. u du n 1 5. eu du eu C 1 6. du ln u C u n Definition: n 1 n 1 Differentials If y f ( x) defines a differentiable function, then 1. The differential dx of the independent variable x is an arbitrary real number. 2. The differential dy of the dependent variable y is defined as the product of f ‘( x) and dx: dy f ‘( x)dx Procedure: Integration by Substitution 1. Select a substitution that appears to simplify the integrand. In particular, try to select u so the du is a factor in the integrand. 2. Express the integrand entirely in terms of u and du, completely eliminating the original variable and its differential. 3. Evaluate the new integral if possible. 4. Express the antiderivative found in step 3 in terms of the original variable. 169 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 In Problems 1–8, find each indefinite integral and check the result by differentiating. 1. ∫ (5 x3 + 7 x 2 − 5) 4 (15 x 2 + 14 x) dx 2. ∫ 4 (5 x5 − 7 x + 3)(25 x 4 − 7) dx 170 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 3. ∫ 3t 3 + 4 6t 4 + 32t − 7 Chapter 5 dt 4. ∫ e −1.5x dx 171 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 5. x( x 7)7 dx 6. 2(ln(3 x 2 )) 4 dx x 172 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 7. 8. 1 x e 6 1 5 x Chapter 5 dx dy 12 x 2 (2 x3 7)5 dx 173 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 9. The indefinite integral can be found in more than one way. Given the integral, 2 2 2 x( x 3) dx, first use the substitution method to find the indefinite integral and then find it without using substitution. 174 Copyright © 2019 Pearson Education, Inc.

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