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

Method: Fundamental Theorem of Calculus Pt. 1

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

a x f ( t ) dt

  • If the bounds need to be flipped you will need to use the “ pull the negative antiderivative rewrite to flip the upper bound and lower bound.

x a f ( t ) dt = a x f ( t ) dt

  • If both bounds are an equation, you will need to split the integral into two integrals being added . You very often will want to split the integral at x =0.

sin ( x ) sin ( x ) f ( t ) dt = sin ( x ) 0 f ( t ) dt + 0 sin ( x ) f ( t ) 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 ) .

i)                     F ( x ) = a x f ( t ) dt

ii)                   d dx F ( x ) = d dx a x f ( t ) dt

iii)                 F ( x ) = d dx a x f ( t ) dt = f ( x )

iv)                 F ( x ) = f ( x )

You really want to be able to get from (i) to (iv) without applying (ii) and (iii). Those middle steps are there just to show you how one truly gets from (i) to (iv).

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.

d dx a sin ( x ) f ( t ) dt = f ( sin ( x ) ) cos ( x )

Since the upper bound in this situation is more than just the standard, x , you are essentially performing a Chain Rule . This means you will need to multiply your result from Step 2, f ( sin ( x ) ) , by the derivative of the upper bound. In this example the derivative of sin(x) would be cos(x) . You would multiply by cos(x) .

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