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 2: Fundamental Theorem of Calculus Pt. 1

Compute the derivative:

d dx sin ( x ) 7 t 1 + t 2 dt

Step 1: Ensure that the lower bound of your antiderivative is a constant, a , and the upper bound is an equation, x .

 

Here the bounds are not in the proper order. You will need to “pull a negative” to flip the order of the bounds.

sin ( x ) 7 t 1 + t 2 dt

 

7 sin ( x ) t 1 + t 2 dt

Step 2: Take the derivative of your antiderivative.

At this point you essentially cancel out the antiderivative and plug your upper bound equation , x , into the equation that was inside the antiderivative, f ( t ) .

 

Here you would replace all the t ’s in your equation with sin(x) ’s, the upper bound equation.

d dx   7 sin ( x ) t 1 + t 2 dt = ( sin ( x ) ) 1 + ( sin ( x ) ) 2

Step 3 (only if your equation is more than just an x ): If the upper bound is more than just a basic x , then you will need to multiply the equation you created in Step 2 , f ( x ) , by the derivative of the upper bound.


Here your upper bound is more than just an x , the upper bound is the equation sin(x) . You will need to multiply your result from Step 2 by the derivative of sin(x) , which is cos(x) .

d dx   7 sin ( x ) t 1 + t 2 dt = ( sin ( x ) ) 1 + ( sin ( x ) ) 2 cos ( x )

Final Result:

The derivative of the equation F ( x ) = sin ( x ) 7 t 1 + t 2 dt would be F ( x ) = ( sin ( x ) ) 1 + ( sin ( x ) ) 2 cos ( x ) .

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