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^(x) Definite Integral

3 7 e ( 9 x )  dx

Step 1: Simplify and look for algebraic rewrites.

None in this example.

3 7 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 ) .

3 7 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

3 7 e ( 9 x )  dx = 1 9 e ( 9 x ) | x = 3 x = 7

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

 

3 7 e ( 9 x )  dx = 1 9 e ( 9 x ) | x = 3 x = 7

= ( 1 9 e ( 9 ( 7 ) ) ) ( 1 9 e ( 9 ( 3 ) ) ) 2 . 54 × 10 26

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 ) = e ( 9 x ) and the x-axis on the x-interval [ 3 , 7 ] is 2 . 54 × 10 26 .

 

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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