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Basic Laws of Integration

Let f(x) and g(x) be any two functions of x, then

  1. Description: 82234.png
     
    where c is any constant
     
     
    Example
    Description: 82249.png
    Solution
    Description: 82255.png
     
    Description: 82261.png 
  2. Description: 82267.png
     
     
    Example
    Description: 82282.png
    Solution
    Given Description: 82294.png
     
    Here ee is a constant.
     
    Description: 82367.png  

Some More Standard Results

1.

Description: 82496.png 

Description: 82504.png 

2.

Description: 82512.png 

Description: 82526.png 

3.

Description: 82535.png 

Description: 82541.png 

4.

Description: 82547.png 

Description: 82553.png 

5.

Description: 82563.png 

Description: 82628.png 

6.

Description: 82642.png 

Description: 82653.png 

7.

Description: 92454.png 

Description: 82671.png 

8.

Description: 82677.png 

Description: 82684.png 

9.

Description: 82690.png 

Description: 82698.png 

10.

Description: 82704.png 

Description: 82770.png 

11.

Description: 82781.png 

Description: 82791.png 

12.

Description: 82804.png 

Description: 82810.png 

 

Note: elog f (xf (x)

 

Example
Evaluate Description: 82824.png
Solution
We know that Description: 82846.png
Description: 82853.png 

 

Example
Evaluate Description: 82859.png
Solution
We know thatDescription: 82866.png
Description: 82932.png 

 

Example
Evaluate Description: 82938.png
Solution
Description: 82947.png
Description: 82959.png 
Description: 82970.png 

 

Example
Description: 83027.png
Solution
Description: 83040.png 
Description: 83046.png 
Description: 83052.png 

Integration by Substitution

Let Description: 83117.png be a given integral. Sometimes it is not possible to be able to integrate f (x) directly. We can substitute f (x) into some other function g(t) to make it readily integrable.

 

 

Let f (x) = g(t) be another function.

 

Description: 83130.png 

Description: 83136.png 
 

Example
Description: 83143.png
Solution
We can see that 1/x is the derivative of log x. So, we can easily use substitution here.
Let log x = t
Differentiating both sides with respect to x, we get
Description: 83151.png 
Description: 83162.png 
Description: 83168.png 
Description: 83179.png 





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