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: e^(ax) Indefinite Integral

e ( 9 x )  dx

 

Step 1: Simplify and look for algebraic rewrites.

None in this example.

e ( 9 x )  dx

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

Here you have 1 chunk, and it is the base case e ( n x ) .

e ( 9 x )  dx

Step 3: Take the antiderivativeof the e ( n x ) special cases using their specific Recipe.

Chunk 1: e ( 9 x )

n = 9

e ( 9 x )  dx = 1 9 e ( 9 x ) + C

DO NOT FORGET THE +C

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 antiderivative of the equation f ( x ) = e ( 9 x ) is the family of graphs f ( x ) = 1 9 e ( 9 x ) + C .

 

All graphs of the form f ( x ) = 1 9 e ( 9 x ) + C have the same derivative (instantaneous rate of change), f ( x ) = e ( 9 x ) , at any x-value .

The only difference between any of original graphs, f ( x ) = 1 9 e ( 9 x ) + C , is just a vertical shift of +C .

In the graph above are examples of 3-different C-values. You can see the instantaneous rate of change ( slope ) is the same at every x-value between all the different examples, no matter the +C .

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