Foundations — Trigonometric integrals — sinᵐ·cosⁿ cases, tan and sec cases

Before you can do that move, you must recognise every symbol on sight. This page builds each one from zero, in the order they depend on each other. Nothing here is assumed. Throughout, "the parent topic" means the main note this page hangs under — Trigonometric integrals — which uses every symbol we build here.

1. Angles, and what $\sin$ / $\cos$ actually are

Look at the figure. The horizontal coordinate of the point is $\cos x$ (the coral segment); the vertical coordinate is $\sin x$ (the mint segment). That is the entire meaning of these two words:

• ==$\cos x$== = "how far right/left" you are on the circle.
• ==$\sin x$== = "how far up/down" you are on the circle.

Because the point sits on a circle of radius $1$, the Pythagoras theorem on the little right triangle (horizontal leg $\cos x$, vertical leg $\sin x$, hypotenuse $1$) reads:

${\cos }^{2}x+{\sin }^{2}x=1.$

That is the first Pythagorean identity — and you just saw why it is true: it is Pythagoras on the radius-1 triangle. (More in Pythagorean identities.)

2. Powers and notation: ${\sin }^{2}x$, ${\sin }^{m}x$

• ${\sin }^{3}x=(\sin x)\cdot (\sin x)\cdot (\sin x)$.
• The letters ==$m$ and $n$== in ${\sin }^{m}x{\cos }^{n}x$ are just placeholders for whole numbers — "some power of sine times some power of cosine". Writing $m$ and $n$ lets us state one rule that covers ${\sin }^2{\cos }^3$, ${\sin }^4{\cos }^4$, and every other combination at once.

3. Even vs odd — the word the whole topic hinges on

Why does a calculus page care? Because of this single fact:

4. The derivative — a slope machine

In the figure, the lavender curve is $\sin x$. At each point the coral tangent line's steepness is the value of $\cos x$, plotted as the mint curve. Where $\sin x$ is climbing fastest (at $x=0$), $\cos x=1$; where $\sin x$ is flat at its peak, $\cos x=0$. That is the picture behind the first two anchors:

Why the minus on $\cos$? Just past $x=0$, the point on the circle is moving upward but leftward — its horizontal coordinate $\cos x$ is shrinking, so its rate of change is negative. That sign is the whole reason $u=\cos x$ substitutions carry a $-du$ later.

Why the topic needs these: substitution (§7) hunts for a chunk of the integrand that is the derivative of another chunk. These facts tell you exactly what $\sin$ and $\cos$'s "derivative-shape" looks like, so you know which brick to peel. The $\tan$ and $\sec$ anchors need those functions first — they arrive in §5.

5. $\tan$ and $\sec$ — two more circle words

Now that $\tan$ and $\sec$ exist, we can state their derivatives — and, crucially, why:

Why these shapes? Apply the quotient rule to $\tan x=\frac{\sin x}{\cos x}$: $\frac{d}{dx}\frac{\sin x}{\cos x}=\frac{\cos x\cdot \cos x-\sin x\cdot (-\sin x)}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}={\sec }^{2}x.$ The last step used the first Pythagorean identity in the numerator. Similarly $\sec x=(\cos x{)}^{-1}$ differentiates to $\frac{\sin x}{{\cos }^{2}x}=\frac{1}{\cos x}\cdot \frac{\sin x}{\cos x}=\sec x\tan x$. So these anchors are derived, not memorised.

Dividing the identity ${\sin }^{2}x+{\cos }^{2}x=1$ through by ${\cos }^{2}x$ gives the second Pythagorean identity. We are allowed to divide only where ==$\cos x\ne 0$== (you can never divide by zero); at the angles where $\cos x=0$, $\tan x$ and $\sec x$ are undefined anyway, so the identity is stated on exactly the domain where its symbols make sense:

$\frac{{\sin }^{2}x}{{\cos }^{2}x}+\frac{{\cos }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}⟹{\tan }^{2}x+1={\sec }^{2}x.$

Why the topic needs it: it is the $\tan$/$\sec$ world's version of "rebuild the leftover" — the counterpart to ${\cos }^2=1-{\sin }^2$.

6. The integral sign $\int$ and $dx$

The figure shows integration as area under a curve: ${\int }_a^{b}fdx$ is the shaded region. But for our topic the useful reading is the "undo a derivative" one.

7. $u$-substitution — the engine

The schematic below traces this exact move on one example — read it top to bottom. The lavender box is the starting integral. The coral arrow peels the derivative-shaped factor $\cos xdx$ and renames it $du$; the mint arrow rebuilds the leftover ${\cos }^{2}x$ as $1-u^2$ using the first Pythagorean identity. Both streams pour into the butter box: a plain polynomial integral in $u$.

8. Absolute value $|\cdot |$ and the logarithm $\ln$

That is why $\int \tan xdx=-\ln |\cos x|+C$ wears bars: $\cos x$ dips negative in quadrants II and III, and $|\cos x|$ keeps the log defined everywhere.

Prerequisite map

Equipment checklist

Connections

• Parent topic — this page feeds directly into it.
• u-substitution — the engine assembled in §7.
• Pythagorean identities — the identities of §1 and §5.
• Power-reduction & double-angle formulas — the both-even backup tool.
• Trigonometric substitution — the reverse idea.
• Reduction formulas — recursion for high powers.
• Integration by parts — alternative route for $\int {\sec }^{3}xdx$.