Fundamental Theorem of Calculus Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Fundamental Theorem of Calculus.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.

Integration undoes differentiation. They're two sides of the same coin.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The Fundamental Theorem links the two operations: the derivative of an accumulation function gives back the integrand, and a definite integral equals the change in any antiderivative.

Common stuck point: The procedure for fundamental theorem of calculus is the easy part; the trap is mixing up the parts. Asking "Am I connecting a definite integral to an antiderivative (F(b)−F(a)F(b)-F(a)) or differentiating an accumulation function back to its integrand?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I connecting a definite integral to an antiderivative (F(b)−F(a)F(b)-F(a)) or differentiating an accumulation function back to its integrand?

Worked Examples

Example 1

easy
Let G(x)=∫0x(t2+1) dtG(x) = \int_0^x (t^2 + 1)\,dt. Find G′(x)G'(x) using FTC Part 1.

Answer

G′(x)=x2+1G'(x) = x^2 + 1

First step

1
FTC Part 1 states: if G(x)=∫axf(t) dtG(x) = \int_a^x f(t)\,dt, then G′(x)=f(x)G'(x) = f(x).

Full solution

  1. 2
    Here f(t)=t2+1f(t) = t^2 + 1, so G′(x)=f(x)=x2+1G'(x) = f(x) = x^2 + 1.
  2. 3
    No integration is needed — the derivative of an integral with variable upper limit is just the integrand evaluated at xx.
FTC Part 1 says differentiation undoes integration when the upper limit is the variable. You simply replace tt with xx in the integrand. This is the key link showing derivatives and integrals are inverse operations.

Example 2

hard
Let H(x)=∫1x2cos⁡t dtH(x) = \int_1^{x^2} \cos t\,dt. Find H′(x)H'(x).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Use FTC Part 2 to evaluate ∫1e1t dt\int_1^e \frac{1}{t}\,dt.

Example 2

medium
If F(x)=∫0xet2 dtF(x) = \int_0^x e^{t^2}\,dt, find F′(x)F'(x).

Example 3

easy
Use the FTC to evaluate ∫032x dx\int_0^3 2x \, dx.

Example 4

easy
If F(x)=∫0xt2 dtF(x) = \int_0^x t^2 \, dt, find F′(x)F'(x).

Example 5

easy
If G(x)=∫2xcos⁡t dtG(x) = \int_2^x \cos t \, dt, find G′(x)G'(x).

Example 6

easy
Use the FTC to evaluate ∫124x3 dx\int_1^2 4x^3 \, dx.

Example 7

easy
If H(x)=∫0xet dtH(x) = \int_0^x e^t \, dt, find H′(x)H'(x).

Example 8

easy
Use the FTC to evaluate ∫0πcos⁡x dx\int_0^{\pi} \cos x \, dx.

Example 9

easy
If F(x)=∫5x(3t+1) dtF(x) = \int_5^x (3t + 1) \, dt, find F′(2)F'(2).

Example 10

easy
Use the FTC to evaluate ∫1e1x dx\int_1^e \frac{1}{x} \, dx.

Example 11

medium
If F(x)=∫0x2sin⁡t dtF(x) = \int_0^{x^2} \sin t \, dt, find F′(x)F'(x).

Example 12

medium
Use the FTC to evaluate ∫−12(2x+3) dx\int_{-1}^{2} (2x + 3) \, dx.

Example 13

medium
If F(x)=∫x4t2 dtF(x) = \int_x^4 t^2 \, dt, find F′(x)F'(x).

Example 14

medium
Use the FTC to evaluate ∫02(ex−1) dx\int_0^2 (e^x - 1) \, dx.

Example 15

medium
Distinguish: which is FTC Part 1 vs Part 2 — (a) ddx∫axf dt=f(x)\frac{d}{dx}\int_a^x f\,dt = f(x), (b) ∫abf=F(b)−F(a)\int_a^b f = F(b)-F(a)?

Example 16

medium
If F(x)=∫1x1t dtF(x) = \int_1^x \frac{1}{t}\,dt, find F′(x)F'(x) and F(1)F(1).

Example 17

medium
Use the FTC to evaluate ∫04x dx\int_0^4 \sqrt{x}\,dx.

Example 18

challenge
If F(x)=∫xx2t dtF(x) = \int_{x}^{x^2} t \, dt, find F′(x)F'(x).

Example 19

challenge
A particle has velocity v(t)=3t2−2v(t) = 3t^2 - 2. Find the net displacement from t=0t=0 to t=2t=2.

Example 20

challenge
If ddx∫0sin⁡xet2 dt=?\frac{d}{dx}\int_0^{\sin x} e^{t^2}\,dt = ?, express the derivative.

Example 21

medium
Use the FTC to evaluate ∫02(3x2−4x+1) dx\int_0^2 (3x^2 - 4x + 1) \, dx.

Example 22

medium
If F(x)=∫13xt2 dtF(x) = \int_1^{3x} t^2 \, dt, find F′(x)F'(x).

Example 23

easy
Evaluate ∫023x2 dx\int_0^2 3x^2 \, dx.

Example 24

easy
Evaluate ∫0π/2cos⁡x dx\int_0^{\pi/2} \cos x \, dx.

Example 25

easy
Evaluate ∫131x2 dx\int_1^3 \frac{1}{x^2} \, dx.

Example 26

easy
Evaluate ∫01ex dx\int_0^1 e^x \, dx.

Example 27

easy
Evaluate ∫−11x3 dx\int_{-1}^{1} x^3 \, dx.

Example 28

easy
Evaluate ∫04x dx\int_0^4 \sqrt{x} \, dx.

Example 29

medium
If F(x)=∫1x31t dtF(x) = \int_1^{x^3} \frac{1}{t} \, dt, find F′(x)F'(x).

Example 30

medium
Evaluate ∫0πsin⁡x dx\int_0^{\pi} \sin x \, dx.

Example 31

medium
If F(x)=∫2x5(t2+1) dtF(x) = \int_{2x}^{5} (t^2 + 1) \, dt, find F′(x)F'(x).

Example 32

medium
Evaluate ∫142x−1x dx\int_1^4 \frac{2x - 1}{\sqrt{x}} \, dx.

Example 33

medium
A car's velocity is v(t)=2t+1v(t) = 2t + 1 m/s. Use FTC to find total distance traveled from t=0t = 0 to t=4t = 4 s.

Example 34

medium
If F(x)=∫0xte−t2 dtF(x) = \int_0^x t e^{-t^2} \, dt, find F′(x)F'(x).

Example 35

medium
Evaluate ∫0ln⁡22e2x dx\int_0^{\ln 2} 2 e^{2x} \, dx.

Example 36

medium
State and apply: if ff is continuous and ∫2xf(t)dt=x2−4\int_2^x f(t) dt = x^2 - 4, find f(x)f(x).

Example 37

hard
Find ddx∫xx2sin⁡t dt\frac{d}{dx} \int_{x}^{x^2} \sin t \, dt.

Example 38

hard
Evaluate ∫0π/4sec⁡2x dx\int_0^{\pi/4} \sec^2 x \, dx.

Example 39

hard
A water tank's volume is V(t)=50+∫0t(6−r)drV(t) = 50 + \int_0^t (6 - r) dr. Find V′(t)V'(t) and the maximum volume.

Example 40

hard
Evaluate ∫012x1+x2 dx\int_0^1 \frac{2x}{1 + x^2} \, dx.

Example 41

hard
Evaluate ∫12ln⁡x dx\int_1^2 \ln x \, dx.

Example 42

hard
If F(x)=∫0cos⁡xt2 dtF(x) = \int_0^{\cos x} t^2 \, dt, find F′(x)F'(x).

Example 43

challenge
Use FTC to find the average value of f(x)=x2f(x) = x^2 on [0,3][0, 3].

Example 44

challenge
Let G(x)=∫0x(t−1)(t−3) dtG(x) = \int_0^x (t - 1)(t - 3) \, dt. Find all x>0x > 0 where GG has a local minimum.
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Related Concepts

DerivativeIntegral

Background Knowledge

These ideas may be useful before you work through the harder examples.

derivativeintegral