Calculus 1-More Properties of Integrals, guaranteed 100% Pass

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More Properties of Integrals


Integrating Even and Odd Functions:

Def. A function is even if 𝑓(−𝑥) = 𝑓(𝑥). A function is odd if
𝑓(−𝑥) = −𝑓(𝑥).
Odd Function
Even Function
𝑓(𝑥) = 𝑥 3

𝑓(𝑥) = 𝑥 2




Ex. Determine whether each of the functions is even, odd, or neither.

a. 𝑓(𝑥 ) = 𝑥 5 + 2𝑥 3
b. 𝑓(𝑥 ) = 𝑥 5 + 2𝑥 3 + 1
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c. 𝑓(𝑥 ) =
𝑥 4 +2𝑥 2 +1
d. 𝑓(𝑥 ) = 𝑠𝑖𝑛3 𝑥
e. 𝑓(𝑥 ) = 𝑐𝑜𝑠 3 𝑥


a. 𝑓(−𝑥 ) = (−𝑥 )5 + 2(−𝑥 )3 = −𝑥 5 − 2𝑥 3 = −𝑓(𝑥 ); odd.
b. 𝑓(−𝑥 ) = (−𝑥 )5 + 2(−𝑥 )3 + 1 = −𝑥 5 − 2𝑥 3 + 1; neither.
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c. 𝑓 (−𝑥 ) = = 𝑥 4 +2𝑥 2 +1 = 𝑓(𝑥); even.
(−𝑥)4 +2(−𝑥)2 +1

d. 𝑓(−𝑥 ) = 𝑠𝑖𝑛3 (−𝑥 ) = (sin(−𝑥 ))3 = (−𝑠𝑖𝑛𝑥 )3
= − sin3 𝑥 = −𝑓(𝑥); odd.
e. 𝑓(−𝑥 ) = 𝑐𝑜𝑠 3 (−𝑥 ) = (cos(−𝑥 ))3 = (𝑐𝑜𝑠𝑥 )3 = 𝑓(𝑥 ); even.

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Theorem: Let 𝑎 be a positive real number and let 𝑓 be an integrable function on
the interval [−𝑎, 𝑎].
𝑎 𝑎
If 𝑓 is even then ∫−𝑎 𝑓 (𝑥 )𝑑𝑥 = 2 ∫0 𝑓 (𝑥 )𝑑𝑥
𝑎
If 𝑓 is odd then ∫−𝑎 𝑓 (𝑥 )𝑑𝑥 = 0.



𝑓(𝑥) Even 𝑓(𝑥) odd


−𝑎
𝑎




−𝑎 𝑎

Ex. Evaluate the following definite integrals using symmetry (𝑓 odd/even).
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a. ∫−2(2𝑥 3 − 3𝑥 2 )𝑑𝑥
𝜋
b. ∫ (2𝑠𝑖𝑛3 𝑥 − 3𝑐𝑜𝑠𝑥 )𝑑𝑥
2
𝜋
−2
3 sin (4𝑥)
c. ∫−3 𝑑𝑥
𝑥 6 +3
2
d. ∫−2(3 + |𝑥|)𝑑𝑥


2 2 2
a. ∫−2(2𝑥 3 − 3𝑥 2 )𝑑𝑥=2 ∫−2 𝑥 3 𝑑𝑥 − 3 ∫−2 𝑥 2 𝑑𝑥
𝑥 3 is odd and 𝑥 2 is even so
2
= 0 − 3(2) ∫0 𝑥 2 𝑑𝑥
1 2
= 6 ( 𝑥 3 ) ⃒ = 2(23 − 03 ) = 16.
3 0