Basic integration rules — power, trig, exponential, log: Foundations

Before you can enjoy "read the derivative rules backwards", you must be certain about what each squiggle on the page means. Below, every symbol is unpacked: plain words → the picture → why the topic needs it. Read top to bottom; nothing is used before it is built.

1. Function $f (x)$ — a machine that turns numbers into numbers

The picture is a graph: for every horizontal position $x$ , the curve sits at a height $f (x)$ . That height is all a function is — a height that depends on where you stand.

2. The variable $x$ and $d x$ — "which knob am I turning?"

Imagine a slider you can push left and right along the horizontal axis. That slider is $x$ . The symbol $d x$ is a nudge of that slider — a tiny step to the right.

3. Slope and the derivative $\frac{d}{d x}$ — the forward machine

For a curve, the slope changes as you move. At each point we draw the tangent line — the straight line that just kisses the curve there — and read its slope.

See Derivatives — basic rules for the forward rules; every integral rule is one of those flipped.

4. The prime notation $f^{'} (x)$ vs $\frac{d}{d x}$ — same idea, two costumes

The parent note writes both — e.g. " $F^{'} (x) = f (x)$ " and " $\frac{d}{d x} sin x = cos x$ ". They are interchangeable. Use $\frac{d}{d x}$ when you want to be loud about which variable; use $f^{'}$ when it's obvious.

5. The integral sign $\int$ and $+ C$

An antiderivative $F$ is any function with $F^{'} (x) = f (x)$ . Because of the shift-family, there are infinitely many — that is precisely what $+ C$ captures.

6. Exponents you must be fluent in

7. Absolute value $∣ x ∣$ and the special number $e$ , and $ln$

The log rule is $\int \frac{1}{x} d x = ln ∣ x ∣ + C$ . The bars appear because $\frac{1}{x}$ makes sense for negative $x$ , but the plain $ln x$ does not — so $∣ x ∣$ patches the domain to all $x \neq = 0$ .

8. The core trig functions $sin, cos, tan, sec$ …

9. Linearity — split and scale

Prerequisite map

Every arrow feeds the same destination: the parent topic is just these foundations, read in reverse, with a $+ C$ tacked on.

Equipment checklist

What does

f (x)

mean as a picture?

The height of the curve above the point

x

on the horizontal axis.

What is the derivative

f^{'} (x)

in one sentence?

A new function giving the slope of the tangent line to

f

at each

x

Are

f^{'} (x)

and

\frac{d}{d x} f (x)

the same?

Yes — identical meaning, two notations for the slope-function.

Why does every indefinite integral end in

+ C

Slope fixes only the shape, not the vertical height; differentiation flattens any constant to 0, so we can't recover it.

Rewrite

x

\frac{4}{x ^{3}}

, and

\frac{6}{x}

as powers.

x^{1/2}

4 x^{- 3}

6 x^{- 1}

What does a negative exponent

x^{- k}

mean?

The reciprocal,

\frac{1}{x ^{k}}

Why does the log integral use

∣ x ∣

not

x

\frac{1}{x}

is defined for negative

x

too;

∣ x ∣

extends the antiderivative to all

x \neq = 0

What is

ln x

asking?

e

to what power gives

x

?" — it undoes

e^{x}

On the unit circle, which coordinate is

cos x

The horizontal coordinate of the moving point.

Which trig slope carries a minus sign?

\frac{d}{d x} cos x = - sin x

— the source of the

\int sin x d x = - cos x

trap.

State the two freedoms linearity gives you.

Split a sum into separate integrals; pull constant multipliers outside.

Is there a product rule for integration?

No — use integration by parts or substitution instead.