UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
2.3 Formula
∫ | |
The above formula states that the integral of a fraction having its numerator the differential of its
denominator equals the natural logarithm of the denominator.
( )
Example 6. Evaluate ∫
Solution: We set ( ) ( ).
Take note that the numerator of the integrand requires only the constant factor 3 to complete the
exact differential of the numerator. Hence,
( ) ( )
∫ ∫
| |
⁄
Example 7. Evaluate ∫
Solution: Let ( ) . The exact differential of the
denominator needs constant 10. Hence,
⁄ ⁄
∫ ∫
⁄
| |
[ | | | |]
[ | ( )| | |]
[ | | ] [ ]
( )
Example 8. Evaluate ∫ .
Solution: The integrand is an improper fraction. In case like this, we divide the numerator by the
denominator to be able to use Formula to integrate the remainder over the denominator.
( )
∫ ∫ ( ) ∫ * +
Integral Calculus Module 5 – Integration of Logarithmic and Exponential Functions Page 41
UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
* | |+
[ ] [ ] *** Recal:
( )
Example 9.Evaluate ∫
Solution: We transform the integrand to a form integrable as follows:
∫ ∫ ∫
The first term is integrable using Formula ∫ with . We need to
intoduce the constant factor of before the integral sign to complete the exact differential of
the numerator. On the other hand, the second term is reduced to a form integrable using Formula
with
∫ | | ∫
| | ∫( )
( )
| | ( )
| |
| | | |
Example 10.Evaluate ∫ .
Solution: Let ( ) .
∫ ∫
[ | |]
[ | |] [ | |]
[ | | | |] ***Recall Property of logarithm:
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UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
[ | ( )| ]
[ ]
2.4 Formulas
Since the Naperian constant e (2.71828…) is the principal base of Calculus, Formula is more
often used than Formula . Observe that in both formulas, u is an exponent. This makes the use
of the formulas different on that of the Power formula wherein the exponent is a constant.
Example 11. Evaluate∫( )
Solution: we consider ( ) , then, the given
expression falls under Formula There is a need to introduce the constant factor 3 inside and
outside the integral sign to complete the needed .
∫( ) ∫ ( )
( )
Example 12. Evaluate∫
Solution: The given integrand needs to be reduced to integrable form by first expanding square of
the binomial on the numerator, then, dividing the result by the denominator. Thus,
( )
∫ ∫( )
∫( )
∫( ) *** Use ∫ , with , then, introduce
factor
( )
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UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
Example 13. Evaluate∫ √
Solution: Notice that Formula will not work to evaluate the given integral.The presence of the
exponential function does not imply that Formula can do the task. However, a closer look
tells that the Power formula ∫ is an effective tool to do the integration process
with the ( ). Introduce constant factor outside the integral sign
to neutralize the effect of introducing complete the needed
( )
∫ √
√( )
( )√
√
√
Example 14. Evaluate∫
√
Solution: We set √ ( ) . Then, apply Formula
√ √
√
√ √ √
∫ ∫ * +
√ √
√
√
* +
√ √
( ) ( )( )
Example 15.Evaluate∫ ( )
Solution: It is apparent that Formula applies to perform the task of evaluating the given integral.
Let ( ) . We complete the needed
by introducing the reciprocal of the missing constant , that is , before the integral sign.
Thus,
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UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
Example 16.Evaluate∫
Solution: First apply the Rule on exponent ( ) to reduce the given to integrable form.
Then, use Formula
∫ ∫( ) ∫( ) ( )
( )
( )
( )
( )
Note: . Hence,
Example 17.Evaluate∫
Solution: Technique is done to bring the given derivative into an integrable form. Divide the
numerator by the denominator, that is, . The ∫ is evaluated using Formula
∫ ∫ ( )
[ ( )]
[ ( )] [ ( )]
[ ( )] [ ( )]
[ ( )]
[ ( ) ]
( )
( )
Alternative Method: ∫
The given differential is multiplied and divided by in order to bring it to an integrable form.
Let , . Hence, ∫ [ | |] | |
| | [ |( )| | |]
[ | | |( )|] [ | |]
Integral Calculus Module 5 – Integration of Logarithmic and Exponential Functions Page 45
UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
SAQ6
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
Evaluate the following integrals using formulas
( )
∫ ∫
∫ ∫
∫ ∫
∫ ∫
( )( )
∫ ∫
Integral Calculus Module 5 – Integration of Logarithmic and Exponential Functions Page 46
UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
∫ ∫
∫
∫
√
∫ ∫
√
∫ ∫
∫
∫
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UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
Activity 2.2
NAME: ____________________________________________________ SCORE: ______________
SECTION: ___________DATE: _______________ PROF: __________________________________
⁄
( )
1. ∫ ( )
6. ∫
⁄
2. ∫ ⁄ 7. ∫
3. ∫ ( ) 8. ∫ ⁄ ( )
⁄ √
4. ∫ ⁄ 9. ∫
5. ∫ 10. ∫
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UNIT 2 – FUNDAMENTAL INTEGRATION FORMULAS
ANSWERS TO SAQ6
1. | |
2. | |
3. | |
4. | |
5. | |
6. | |
7. | |
8. [ | |]
9. | |
10. | |
11. | |
12.
13. √
14.
15.
16. | |
⁄
17.
18.
19.
20.
ANSWERS TO ACTIVITY 2.2
1. 6. ( )
2. √ 7.
3. 8.
4. ( √ ) 9.
5. ( )( )( ) 10. [ ]
Integral Calculus Module 5 – Integration of Logarithmic and Exponential Functions Page 49
