Understanding Definite Integrals and Their Properties
Explore the concept of definite integrals, rules satisfied by them, and examples of evaluating integrals. Learn about the symbol notation, theorem related to antiderivatives, and properties such as zero width interval, constant multiple, sum, difference, and additivity. Practice solving integral problems involving trigonometric and exponential functions through detailed explanations and solutions.
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Definite Integral Definite Integral
Definite Integral Rules satisfied by definite integrals Rules satisfied by definite integrals Integration Integration of (du/u) of (du/u) The integral of tan x and cot x The integral of tan x and cot x Integration of Exponential Function ( Integration of Exponential Function (ex) ) Integration of a Integration of au u
Definite Integrals Let (x) be a function defined on a closed interval [a, b]. We say that a number I is the definite integral of over [a, b]. The symbol for the number I in the definition of the definite integral is ? ? = ? ? ?? ? Where
Theorem If is continuous at every point of [a, b] and F is any antiderivative of on [a, b], then ? ? ? ?? = ? ? ?(?) ? The usual notation for F(b)-F(a) is ? ?? ?(?)? ? ?(?)? Rules satisfied by definite integrals When and g are integrable, the definite integral satisfies Rules 1 to 5 below : 1. Order of integration: ? ? ? ? ?? = ? ? ?? ? ?
2. Zero width interval ? ? ? ?? = ? ? 3. Constant Multiple ? ? ?? ? ?? = ? ? ? ?? ? ? ? ? ? ? ?? = ? ? ?? ? ? 4. Sum and difference ? ? ? (? ? ?? ? ? ) ?? = ? ? ?? ? ? ?? ? ? ? 5. Additivity ? ? ? ? ? ?? + ? ? ?? = ? ? ?? ? ? ?
Example Evaluate the following integrals ?cos? ?? 1. 0 Sol. ? ?= sin? sin0 = 0 0 = 0 cos? ?? = sin?0 0 0 2. sec? tan? ?? ? 4 Sol. 0 ? 4 0 sec? tan? ?? = sec? = sec0 sec = 1 2 ? 4 ? 4
4(3 4 ?2)?? 3. 1 ? 2 Sol. 4 4 (3 2 ? 4 3 2+4 3 2+4 3 2+4 ? = [4 4 [1 ?2)?? = 1 = 9 5 = 4 ?1 1 ? 2cot????2? ?? 4. ? 4 Sol. Using substitution Let ? = cot? , ?? = ???2? ??, ?? = ???2? ?? When ? =? 4 ,? = cot? ? 2 ,? = cot? 4= 1 ??? when ? = 2= 0 0 ?2 2 1 ? 2cot????2? ?? = 1 0? ?? = = 0 1 =1 ? 4 2 2
?3 ???2? sin? ?? 5. 0 Sol. Using substitution Let ? = cos? , ?? = sin? ??, ?? = sin? ?? When ? = 0 ,? = cos0 = 1 ??? when ? = ? ,? = cos? = 1 ? 1 3 ???2? sin??? = 3?2 ?? 0 1 1 ?3 3 1 13?2 ?? = 3 = 13 13= 2 = 1
Integration of (du/u) ?1 ln? = ? ??, ? > 0 1 If u is a differentiable function that is never zero, 1 ? ?? = ln|?| + ? Or ? (?) ?(?) ?? = ln ?(?) + ? Example Evaluate the following integral 2 2? ?2 5 ?? 0 Using substitution Let ? = ?2 5,?? = 2? When ? = 0,? = ?2 5 = 5, ??? ? ?? ? = 2, ? = ?2 5 = 1
1?? 1= ln| 1| ln 5 = 0 ln| 5| = ? = ln|?| 5 5 Example Evaluate the following integral ? 2 4cos? 3 + 2sin? ?? ? 2 Using substitution Let ? = 3 + 2sin?,?? = 2cos??? When ? = ? 2,? = 3 + 2sin? = 3 + 2sin( ? 2,? = 3 + 2sin? = 3 + 2sin(? 2) = 3 2 = 1 ? =? 2) = 3 + 2 = 5 52?? 5= 2ln|5| 2ln 1 = 2ln5 = = 2 ln|?|1 ? 1
The integral of tan x and cot x ??? ? ?? = ??? ? ??? ? ?? = ?? = ln ? + ? ? 1 ??? ? ?? = ln|cos? | + ? = ln |cos?|+ ? = ln|sec?| + ? ??? ? ?? = ??? ? ??? ? ?? = ?? ?= ln ? + ? ??? ? ?? = ln sin? + ? = ln|csc? | + ? Example ? 6 ? 3 tan??? 2=1 ? 3 tan2? ?? = 2ln|sec?| 0 0 0 1 2ln|sec?| 0 ? 3=1 ln2 ln1 =1 2ln2 = ln 2 2
Integration of Exponential Function (ex ) ???? = ??+ ? Example: Evaluate the following integral ln 2?3? ?? 1. 0 Sol. Let u= 3x , du = 3 dx ,dx=du/3 ln 2 ln 8 ???? 3=1 ln 8 ?3? ?? = 3 ?? 0 0 0 =1 ?ln 8 ?0 =1 38 1 =7 3 3 ? 2?sin ?cos??? 1. 0 ? 2 ? 2= ?1 ?0= ? 1 ?sin ?cos? ?? = ?sin ? 0 0
Integration of au ?? ln?+ ? ???? = Example 2? ln2+ ? 2??? =
