Mathematics Question

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 n1  C, n  1  x dx  n 1 x x  e dx  e  C 1 x0  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)]n1  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 n1  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.