Maths I — Week 6: Integration Fundamentals

1039 words

5 min read

View

Maths I — Week 6: Integration Fundamentals

The Big Idea: Differentiation breaks functions apart. Integration puts them back together. If you know the speed at every moment, integration gives you the total distance. If you know the marginal cost of each unit, integration gives you the total cost. It is the accumulation machine.

1. The Antiderivative (Indefinite Integral)

F(x)

is called an antiderivative of

f(x)

F'(x) = f(x)

\int f(x)\,dx = F(x) + C

Why $+C$ ? Differentiation erases constants (

\frac{d}{dx}(F(x) + 5) = F'(x)

). So when we reverse that, we must account for any constant that could have been there — hence the family of functions represented by

+C

Omitting

+C

from an indefinite integral is a guaranteed mark loss. Indefinite integrals represent families of functions, not a single one.

2. Standard Integration Table

$f(x)$	$\int f(x)\,dx$
$x^n$ ( $n \neq -1$ )	$\dfrac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln
$e^x$	$e^x + C$
$e^{ax}$	$\frac{1}{a}e^{ax} + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$a^x$	$\frac{a^x}{\ln a} + C$

3. The Fundamental Theorem of Calculus (FTC)

This is the most important theorem in calculus. It bridges differentiation and integration.

FTC Part 1 — Differentiation of an Integral (Derivative recovers the integrand)

\frac{d}{dx}\int_a^x f(t)\,dt = f(x)

Use case: Immediately resolve

\frac{d}{dx}

applied to an integral whose upper limit is

x

Chain Rule Extension: If the upper limit is

g(x)

\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)

FTC Part 2 — Evaluation of Definite Integrals

\int_a^b f(x)\,dx = F(b) - F(a)

Use case: Evaluating any definite integral with a known antiderivative

F(x)

The constant

+C

cancels out in definite integrals:

[F(x)+C]_a^b = F(b)+C - F(a)-C = F(b)-F(a)

4. Properties of Definite Integrals

Property	Formula
Direction reversal	$\int_a^b f = -\int_b^a f$
Zero width	$\int_a^a f = 0$
Linearity	$\int_a^b cf = c\int_a^b f$
Sum	$\int_a^b [f+g] = \int_a^b f + \int_a^b g$
Additivity	$\int_a^b f = \int_a^c f + \int_c^b f$
Even functions	$\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx$
Odd functions	$\int_{-a}^a f(x)\,dx = 0$

5. Signed Area vs. Geometric Area

The definite integral

\int_a^b f(x)\,dx

computes signed area:

Regions above the x-axis contribute positively.
Regions below the x-axis contribute negatively.

For actual (geometric) area, integrate the absolute value:

\int_a^b |f(x)|\,dx

Procedure for Area Enclosed by Curve and X-axis

Find where $f(x) = 0$ (the roots that bound the region).
Determine the sign of $f(x)$ on each sub-interval.
Integrate with appropriate sign adjustments.

Worked Example:

Area enclosed by $y = x^2 - 4$ and the x-axis.

Roots: $x = \pm 2$ .
$f$ is negative on $(-2, 2)$ .
Area $= -\int_{-2}^{2}(x^2-4)\,dx = -\left[\frac{x^3}{3} - 4x\right]_{-2}^{2}$ .
$= -\left[\left(\frac{8}{3}-8\right) - \left(\frac{-8}{3}+8\right)\right] = -\left[\frac{16}{3} - 16\right] = \frac{32}{3}$ .

6. Pattern — Standard Anti-Differentiation

What to recognize: A polynomial or sum of standard functions without a composite inside.

Abstract Solution (Strategy)

[Decompose]: Break the integrand into simpler standard forms.
[Apply Table]: Integrate each term using the table above.
[Add $+C$ ].

Worked Example:

$\int \left(3x^2 + 2e^x - \frac{1}{x}\right)\,dx$

Term 1: $3 \cdot \frac{x^3}{3} = x^3$ .
Term 2: $2e^x$ .
Term 3: $-\ln|x|$ .
Result: $x^3 + 2e^x - \ln|x| + C$ .

7. Pattern — FTC Part 1 Applied

What to recognize:

\frac{d}{dx}

applied to an integral with a variable upper bound.

Worked Example:

$\frac{d}{dx}\int_0^x \sin(t^2)\,dt$

By FTC Part 1 directly: $\sin(x^2)$ .

Chain Rule Extension Example:

$\frac{d}{dx}\int_1^{x^3} e^t\,dt$

$= e^{x^3} \cdot 3x^2$ .

8. Common Mistakes

Mistake	Why it happens	Correct approach
Forgetting $+C$ in indefinite integrals	Treating integration like definite integrals.	Indefinite = always $+C$ . Definite = no $+C$ (it cancels).
Wrong antiderivative for $1/x$	Applying power rule: $x^{-1} \to \frac{x^0}{0}$ (undefined).	$\int \frac{1}{x},dx = \ln
Signed area confusion	Just integrating without checking if $f$ dips below x-axis.	Split the integral at roots; negate below-axis segments for geometric area.
Applying FTC Part 1 when lower limit isn't constant	Misidentifying which part to use.	If both limits are variable, subtract: $\int_g^h f = F(h) - F(g)$ , then differentiate using Chain Rule on each.

9. Flashcards

10. Practice Targets

Compute $\int_1^4 \left(3\sqrt{x} - \frac{2}{x}\right)\,dx$ .
Find the area enclosed between $y = \sin x$ and the x-axis for $x \in [0, 2\pi]$ .
Apply FTC Part 1: find $\frac{d}{dx}\int_1^{x^2} e^{t^2}\,dt$ .
Attempt Graded Assignment 6 Questions 1–5.

11. Connections

Connects to	How
Week 4 — Derivatives	Integration is the reverse of differentiation. Every antiderivative formula is a derivative rule run backwards.
Week 7 — Advanced Integration	Substitution and Integration by Parts extend the standard table to composite functions and products.
Statistics I	Probability density functions (PDFs) integrate to 1 — direct application of definite integrals.

Maths I — Week 6: Integration Fundamentals

1. The Antiderivative (Indefinite Integral)

2. Standard Integration Table

3. The Fundamental Theorem of Calculus (FTC)

FTC Part 1 — Differentiation of an Integral (Derivative recovers the integrand)

FTC Part 2 — Evaluation of Definite Integrals

4. Properties of Definite Integrals

5. Signed Area vs. Geometric Area

Procedure for Area Enclosed by Curve and X-axis

6. Pattern — Standard Anti-Differentiation

Abstract Solution (Strategy)

7. Pattern — FTC Part 1 Applied

8. Common Mistakes

9. Flashcards

10. Practice Targets

11. Connections

Intelligence Hub