Mathfolis

Fundamental Theorem of Calculus

Unit 6 · Integration & Accumulation

The Fundamental Theorem of Calculus

FTC Part 1 says the derivative of an accumulation function equals the integrand. FTC Part 2 evaluates definite integrals via antiderivatives. Together they show that differentiation and integration are inverse processes.

FTC Part 1 (Accumulation Function)

Basic form
ddx∫axf(t) dt=f(x)\frac{d}{dx}\int_a^x f(t)\,dt = f(x)
With chain rule (upper limit g(x))
ddx∫ag(x)f(t) dt=f(g(x))⋅g′(x)\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\cdot g'(x)

FTC Part 2 (Evaluation Theorem)

∫abf(x) dx=F(b)−F(a)where F′=f\int_a^b f(x)\,dx = F(b) - F(a) \quad \text{where } F' = f
Caution: When the upper limit is g(x) (not just x), always multiply by g′(x) via the Chain Rule — the most common FTC error.
Type 1

FTC Part 1 — Derivatives of Accumulation Functions

Differentiate integrals with variable upper limits. Apply the chain rule when the upper limit is a function of x.

Example 1

Find the derivative.

ddx∫2x(t3+sin⁡t) dt\frac{d}{dx}\int_2^x (t^3 + \sin t)\,dt
Example 2

Find the derivative.

ddx∫1x21+t4 dt\frac{d}{dx}\int_1^{x^2} \sqrt{1+t^4}\,dt
Example 3

Find the derivative.

ddx∫2x2t3+1 dt\frac{d}{dx}\int_2^{x^2}\sqrt{t^3+1}\,dt

Practice more of this type— AI-generated · infinite problems

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Type 2

FTC Part 2 — Evaluating Definite Integrals

Find the antiderivative F, then compute F(b) − F(a). The +C cancels and can be omitted.

Example 4

Evaluate the definite integral.

∫02(3x2−4x+1) dx\int_0^2 (3x^2 - 4x + 1)\,dx
Example 5

Evaluate the definite integral.

∫0π/2(cos⁡x+2sin⁡x) dx\int_0^{\pi/2}(\cos x + 2\sin x)\,dx

Practice more of this type— AI-generated · infinite problems

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← AntiderivativesU-Substitution →