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

Example 1: Indefinite Integral Trig Functions

sin ( x )  dx


Step 1: Simplify and look for algebraic rewrites.

None in this example.

sin ( x )  dx

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

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

sin ( x )  dx

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

Chunk 1: sin(x)

sin ( x )  dx = cos ( x ) + 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 ) = sin ( x ) is the family of graphs f ( x ) = cos ( x ) + C .


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

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