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: Indefinite Integral Inverse Trig

1 1 + x 2  dx

 

Step 1: Simplify and look for algebraic rewrites.

None in this example.

1 1 + x 2  dx

Step 2: Identify any term(s) that include one of the six inverse trig function special cases.

Here you have 1 chunk, and it is one of the inverse trig function special cases.

1 1 + x 2  dx

Step 3: Take the antiderivativeof the inversetrig function special cases using their specific Recipe.

 

Chunk 1: 1 1 + x 2

1 1 + x 2  dx = tan 1 ( 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 ) = 1 1 + x 2 is the family of graphs f ( x ) = tan 1 ( x ) + C .

 

All graphs of the form f ( x ) = tan 1 ( x ) + C have the same derivative (instantaneous rate of change), f ( x ) = 1 1 + x 2 , at any x-value .

The only difference between any of original graphs, f ( x ) = tan 1 ( 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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