The Two Main Antiderivative Rules
Special Case Antiderivatives (The ones you have to memorize)
U-Substitution
Initial Value Problems: Finding the +C
Fundamental Theorem of Calculus Part 1

Example 1: Definite Integral ln(x)

3 7 9 x  dx

Step 1: Simplify and look for algebraic rewrites.

 

None in this example.

3 7 9 x  dx

Step 2: Identify any term(s) that includes the base case n x 1 .

 

Here you have 1 chunk, and it is the base case 9 x .

3 7 9 x  dx

Step 3: Take the antiderivativeof the n x 1 special cases using their specific Recipe.

Chunk 1: 9 x

n = 9

3 7 9 x  dx = 9 ln | x | | x = 3 x = 7

Step 4 ( Definite Integral ONLY ): Evaluate the antiderivative result using the TopBottom method.

3 7 7 x  dx = 7 ln | x | | x = 3 x = 7

= ( 7 ln | ( 7 ) | ) ( 9 ln | ( 3 ) | ) 7 . 625

Final Result Meaning: Remember the Definite Integral will always provide you a definite value , and the Indefinite Integral provides you a family of solutions .

The Net Area between the curve f ( x ) = 9 x and the x-axis on the x-interval [ 3 , 7 ] is 7.625 .

 

Since the final result is positive, you know without even seeing the graph that there is more area above the x-axis than below it.

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