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: Initial Value Problem

Given the differential equation: f ( x ) =   x 3 + 5 x 2 7 x + 3

Determine its specific solution given (6,586).

Step 1: Find your general solution using your standard Indefinite Integral antiderivative process.

 

Her you will find the antiderivative of the given equation, f ( x ) =   x 3 + 5 x 2 7 x + 3 , using your standard antiderivative methods. This problem will only require the use of the power rule and the constant rule.

x 3 + 5 x 2 7 x + 3   dx

x 3 + 5 x 2 7 x 1 + 3   dx

= x 3 + 1 4 + 5 x 2 + 1 3 7 x 1 + 1 2 + 3 x + C

= 1 4 x 4 + 5 3 x 3 7 2 x 2 + 3 x + C

f ( x ) = 1 4 x 4 + 5 3 x 3 7 2 x 2 + 3 x + C

Step 2: Plug your given point ( x ,y) into your general solution from Step 1 and solve for the +C value.

 

Here you are given the point (6,586). You will replace all the x ’s with 6 . You will replace the f ( x ) with 586. Remember f ( x ) = y .

 

Simplify the right side of the equals where all the x ’s were plugged in. You will then have a very straightforward algebra problem you will need to solve for C .

f ( x ) = 1 4 x 4 + 5 3 x 3 7 2 x 2 + 3 x + C

586 = 1 4 ( 6 ) 4 + 5 3 ( 6 ) 3 7 2 ( 6 ) 2 + 3 ( 6 ) + C

586 = 324 + 360 126 + 18 + C

586 = 576 + C

586 576 = 576 576 + C

10 = C

Step 3: Plug the C-value you found in Step 2 into the general solution from Step 1 to get the final result a specific solution.

 

f ( x ) = 1 4 x 4 + 5 3 x 3 7 2 x 2 + 3 x + C

f ( x ) = 1 4 x 4 + 5 3 x 3 7 2 x 2 + 3 x + 10

Final Result Meaning:

The derivative, f ( x ) =   x 3 + 5 x 2 7 x + 3 , whose antiderivative, Indefinite Integral, general solution is f ( x ) = 1 4 x 4 + 5 3 x 3 7 2 x 2 + 3 x + C , has the specific solution, f ( x ) = 1 4 x 4 + 5 3 x 3 7 2 x 2 + 3 x + 10 , that includes the given initial value (6,586).

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