AP Calculus AB and BC

FTC Part 1 vs Part 2

Part 1 differentiates an accumulation function and returns the integrand, which is the statement that differentiation undoes integration. Part 2 evaluates a definite integral as an antiderivative at the top bound minus the same antiderivative at the bottom.

FTC Part 1

Use when: You are asked to differentiate an integral whose upper bound contains the variable.

FTC Part 2

Use when: You are asked to compute the numerical value of a definite integral.

Side by side

	Part 1	Part 2
Statement	$\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$	$\int_a^b f(x)\,dx = F(b) - F(a)$
Operation	Differentiate	Evaluate
Answer	A function	A number
Watch for	A chain rule factor on the bound	Continuity across the interval

Part 1 is the reason accumulation functions can be analysed like any other function: the integrand is the derivative. Where $f$ is positive the accumulation increases, and where $f$ crosses from positive to negative the accumulation has a local maximum.

Part 2 is what makes definite integrals computable without ever forming a Riemann sum. It needs the integrand to be continuous on the closed interval, so applying it blindly across a vertical asymptote produces nonsense.

The Part 1 trap

If the upper bound is a function rather than $x$ alone, the chain rule adds a factor. The derivative of $\int_0^{x^2} f(t)\,dt$ is $f(x^2) \cdot 2x$ .

Frequently asked questions

Which part do I use for an accumulation function question?

Part 1, whenever you are differentiating. Use Part 2 when you need the numerical value at a specific input.

In the CED: Unit 6: Integration and Accumulation