Lesson 5 4 a Indefinite Integrals Review Icebreaker

Lesson 5 -4 a Indefinite Integrals Review

Icebreaker Problem 1: 2 3 - 9) dx ( 3 x ∫ = ¾ x 4 – 9 x + c 1 = ¾ 24 – 9(2) + c – (¾ 14 – 9(1) + c) = Find the derivative of the following: 3 x ∫x² (2 t - 7) dt = t² - 7 t + c = (2 t – 7) dt [2(3 x) – 7](3) = (2 t – 7) dt [2(x²) – 7](2 x)

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

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

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

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

Derivatives of Inverse Trigonometric Functions d 1 -1 ---- (sin x) = ------dx √ 1 - x² d -1 -1 ---- (cos x) = -----dx √ 1 - x² d 1 -1 ---- (tan x) = ------dx 1 + x² d -1 -1 ---- (cot x) = ------dx 1 + x² d 1 ---- (sec-1 x) = -------dx x √ x² - 1 d -1 ---- (csc-1 x) = ------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.

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.

Indefinite Integration Review 1. ∫ 2. ∫a 3. ∫ ex dx = 4. ∫ cos(x) dx = 5. 6. sin(x) dx = = ∫ sec 2(x) dx = ∫ csc 2(x) dx =

Indefinite Integration Review 7. 8. 9. ∫ sec(x)tan(x) dx ∫ csc(x)cot(x) dx ∫ 1 ----- dx x² + 1 = = = ∫ 1 ----- dx 1 - x² 11. ∫ 1 ----- dx x 12. ∫ xn dx 10. = = =

Example Problems with TI-89 Evaluating indefinite integral with our calculator: ∫ (-x 2 + 4 x – 3)dx = -⅓x 3 + 2 x 2 – 3 x + C Hit F 3 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 + 4 x – 3, x, ) C is missing from calculator answer

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

More Practice Problems Now use your knowledge of the formulas and integration rules to evaluate the following: 1 ( x - -----) dx x 4. ∫ 5. ∫ (3 sec(x)tan(x) – 2 csc(x)cot(x)) dx 6. ∫ = (2 sec 2(x) + 4 csc 2(x)) dx = =

More Practice Problems Now use your knowledge of the formulas and integration rules to evaluate the following: 7. 8. 9. ∫ 1 (1 - -----) dx x² + 1 = ∫ 3 (------) dx 1 - x² ∫ 4 – x - x² (--------) dx 2 x = =

More Practice Problems Now use your knowledge of the formulas and integration rules to evaluate the following: 10. ∫ 1 - sin²(x) (--------) dx cos²(x) =

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