Mastering Indefinite Integrals - Concepts & Practice

1. Lesson 5-4a Indefinite Integrals Review

2. Icebreaker 2 Problem 1:(3x3 - 9) dx Find the derivative of the following:(2t - 7) dt ∫ = ¾ x4 – 9x + c 1 = ¾ 24 – 9(2) + c – (¾ 14 – 9(1) + c) = 3x ∫ = t² - 7t + c x² = (2t – 7) dt  [2(3x) – 7](3) = (2t – 7) dt  [2(x²) – 7](2x)

3. Objectives • Solve indefinite integrals of algebraic, exponential, logarithmic, and trigonometric functions • Understand the Net Change Theorem • Use integrals to solve distance problems to find the displacement or total distance traveled

4. Vocabulary • Indefinite Integral – is a function or a family of functions • Distance – the total distance traveled by an object between two points in time • Displacement – the net change in position between two points in time

5. Basic Differentiation Rules d ---- (c) = 0 Constant dx d ---- (xⁿ) = nxn-1Power Rule dx d d ---- [cf(x)] = c ---- f(x) Constant Multiple Rule dx dx d ---- (ex) = exNatural Exponent dx d 1 ---- (ln x) = -----Natural Logarithms dx x

6. Trigonometric Functions Differentiation Rules d d ---- (sin x) = cos x ---- (cos x) = –sin x dx dx d d ---- (tan x) = sec² x ---- (cot x) = –csc² x dx dx d d ---- (sec x) = sec x • tan x ---- (csc x) = –csc x • cot x dx dx Hint: The derivative of trig functions (the “co-functions”) that begin with a “c” are negative.

7. Derivatives of Inverse Trigonometric Functions d 1 d -1 ---- (sin-1 x) = ------------ ---- (cos-1 x) = ----------- dx √1 - x² dx √1 - x² d 1 d -1 ---- (tan-1 x) = ------------- ---- (cot-1 x) = ------------- dx 1 + x² dx 1 + x² d 1 d -1 ---- (sec-1 x) = ------------------ (csc-1 x) = ------------- dx x √ x² -1 dx x √ x² - 1 Interesting Note: If f is any one-to-one differentiable function, it can be proved that its inverse function f-1 is also differentiable, except where its tangents are vertical.

8. Other Differentiation Rules Constant to Variable Exponent Rule d ----- [ax] = ax ln a dx This is a simple example of logarithmic differentiation that we will examine in a later problem. Sum and Difference Rules d d d ---- [f(x) +/- g(x)] = ---- f(x) +/- ---- g(x) dx dx dx In words: the derivative can be applied across an addition or subtraction. This is not true for a multiplication or a division as the next two rules demonstrate.

9. Indefinite Integration Review ∫ ∫ ∫ ∫ ∫ ∫ ex dx = cos(x) dx = ax dx = sec2(x) dx = sin(x) dx = csc2(x) dx =

10. Indefinite Integration Review ∫ ∫ ∫ ∫ ∫ ∫ xn dx = sec(x)tan(x) dx = csc(x)cot(x) dx = 1 ---------- dx = x² + 1 1 ---------- dx = 1 - x² 1 ----- dx = x

11. Example Problems with TI-89 ∫ Evaluating indefinite integral with our calculator: Hit F3 select integration; type in function (t²), integrate with respect to (t), lower limit of integration (1), upper limit of integration (x); close ). Type , and differentiate with respect to x and close ). Should look like this: ∫(-x^2 + 4x – 3,x,) (-x2 + 4x – 3)dx = -⅓x3 + 2x2 – 3x + C C is missing from calculator answer

12. More Practice Problems ∫ ∫ Now use your knowledge of the formulas and integration rules to evaluate the following: ∫ (-x2 + 4x – 3)dx = (2x – 1)2 dx = 3 (----- - 1) dx = x²

13. More Practice Problems ∫ ∫ ∫ (2sec2(x) + 4csc2(x)) dx = (3sec(x)tan(x) – 2csc(x)cot(x)) dx = 1 (x - -----) dx = x Now use your knowledge of the formulas and integration rules to evaluate the following:

14. More Practice Problems ∫ ∫ ∫ 4 – x - x² (---------------) dx = 2x 1 (1 - ---------) dx = x² + 1 3 (-----------) dx = 1- x² Now use your knowledge of the formulas and integration rules to evaluate the following:

15. More Practice Problems ∫ 1 - sin²(x) (---------------) dx = cos²(x) Now use your knowledge of the formulas and integration rules to evaluate the following:

16. Summary & Homework • Summary: • Definite Integrals are a number • Evaluated at endpoints of integration • Indefinite Integrals are antiderivatives • Homework: • Day One: pg 411-413: 1, 7, 8, 17, 20, 23, 33, • Day Two: pg 411-413: 59, 62