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Exponential & Logarithmic Integration

This document provides examples and formulas for integrating exponential and logarithmic functions: 1) It gives the basic integration formulas for exponential functions like e^u, a^u, and 1/u as well as logarithmic functions like ln(u). 2) It also lists some important properties and laws for exponential and logarithmic functions. 3) Finally, it works through 5 examples of evaluating integrals involving exponential and logarithmic terms, applying substitution and other integration techniques to solve them.

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Matthew Jordan
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104 views5 pages

Exponential & Logarithmic Integration

This document provides examples and formulas for integrating exponential and logarithmic functions: 1) It gives the basic integration formulas for exponential functions like e^u, a^u, and 1/u as well as logarithmic functions like ln(u). 2) It also lists some important properties and laws for exponential and logarithmic functions. 3) Finally, it works through 5 examples of evaluating integrals involving exponential and logarithmic terms, applying substitution and other integration techniques to solve them.

Uploaded by

Matthew Jordan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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2/16/2023

Integration of Exponential
and Logarithmic Functions

DOLFUS G. MICIANO
CEAFA, Batstate-U

Some Elementary Formulas


(TC7,Leithold)

• ∫ du = u + C
• ∫ a du = au + C
• ∫ [ f(u) ± g(u)]du = ∫ f(u)du ± ∫ g(u)du
n 1
u
• ∫ un du =  C where n ≠ -1
n 1

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2/16/2023

Integration Formulas: Exponential &


Logarithmic Functions
Given a and e are real nos.,

u au
• ∫ =
a+𝐶
• ∫ eu du =e u + 𝐶

• ∫ =∫ -1 = ln|u| + C

• ∫ ln u 𝑑𝑢 = u ln|u| - u + C

Some Exponential & Logarithmic


Properties/Laws
Exponential: Logarithmic:
• am an =am+n • logb (xy) = logb x + logb y
• (am)n =amn
x
• logb
• (ab)n =anbn y = logb x − logb y
am m-n • logb x r = r logb x
•
an =a
1 ln x
-n • logb x =
•
an =a ln b

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EXAMPLES
Ex. 1. Solve the integral ∫ 𝑒 𝑑𝑥
Solution: −𝑑𝑢
= ∫𝑒 change into u variable
𝑙𝑒𝑡 𝑢 = 2 − 5𝑥 5

𝑑𝑢 = −5𝑑𝑥 = ∫ 𝑒 𝑑𝑢
−𝑑𝑢
= 𝑑𝑥
5 = 𝑒 +C
substitute to x variable

= 𝑒 +C

Ex. 2. Evaluate the integral ∫ 10 𝑑𝑥


2𝑑𝑢
Solution:∫ 10 /
𝑑𝑥 = ∫ 10 3 change variable

3𝑥
𝑙𝑒𝑡 𝑢 = = ∫ 10 𝑑𝑢
2
d𝑢 =
=  +C
2𝑑𝑢
𝑑𝑥 = /
3 =  +C back substitution

= +C

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Ex. 3. Evaluate the integral ∫ 𝑒

Solution:

𝑙𝑒𝑡 𝑢 = 𝑥 + 1 =
d𝑢 = =
2𝑑𝑢 = =  +C

=2 +C

Ex. 4. Evaluate the integral ∫ 𝑑𝑥


Perform division
Solution:
= ∫ 𝑒 + 𝑒 𝑑𝑥
= ∫ 𝑒 𝑑𝑥 +∫ 𝑒 𝑑𝑥
𝑙𝑒𝑡 𝑢 = −𝑥
=∫ 𝑒 𝑑𝑥 + ∫ 𝑒 Interchange
𝑑𝑥 the integrals
d𝑢 = −𝑑𝑥
𝑑𝑥 = −𝑑𝑢 = 𝑒 + ∫ 𝑒 (−𝑑𝑢)
= 𝑒 - ∫ 𝑒 𝑑𝑢
=𝑒 -𝑒 +𝐶
=𝑒 -𝑒 +𝐶

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Ex. 5. Evaluate the integral ∫


( )
1 𝑑𝑥
Solution:
(𝑙𝑛 𝑥 ) 𝑥 =∫ 𝑑𝑢

𝑙𝑒𝑡 𝑢 = ln 𝑥 =∫ 𝑢 𝑑𝑢
= −
d𝑢 = Note: ln xr = r ln x
=
𝐿𝑖𝑚𝑖𝑡𝑠: ln 4 = ln 22 = 2 ln 2
𝑖𝑓 𝑥 = 2 ≫ 𝑢 = ln 2
= −
𝑥 = 4 ≫ 𝑢 = ln 4 =−
=
=− −
Multiply the negative

Exercises: Solve the following integrals

1. ∫ 𝑑𝑥

2. ∫ 𝑑𝑥 Hint: logax =
a

3. ∫ 𝑑𝑥

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