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: Antiderivative of a Constant

 

Definite Integral

Indefinite Integral

3 7 5  dx

5  dx

 

Step 1: Simplify and look for algebraic rewrites.

 

None here.

3 7 5  dx

5  dx

Step 2: Identify any term(s) that are just a constant.

The only term in either example is just a constant, 5.

3 7 5  dx

5  dx

Step 3: Take the antiderivativeof the constant using the Recipe: Numbers by themselves, constants, gain a letter back.

3 7 5  dx = 5 x | x = 3 x = 7

5  dx = 5 x + C

DO NOT FORGET THE +C

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

3 7 5  dx = 5 x | x = 3 x = 7 = 5 ( 7 ) 5 ( 3 ) = 20

 

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 ) = 5 and the x-axis on the x-interval [ 3 , 7 ] is 20.

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.

 

 

The antiderivative of the equation f ( x ) = 5 is the family of graphs f ( x ) = 5 x + C .

 

All graphs of the form f ( x ) = 5 x + C have the same derivative (instantaneous rate of change), f ( x ) = 5 , at any x-value .

The only difference between any of original graphs, f ( x ) = 5 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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