QCE Specialist Mathematics - Unit 4 - Integration techniques
Trigonometric Substitution and Log Integrals | QCE Specialist Mathematics
Learn QCE Specialist integration techniques for trigonometric identities, substitution and logarithmic integrals.
Updated 2026-05-18 - 5 min read
QCAA official coverage - Specialist Mathematics 2025 v1.4
Exact syllabus points covered
- Integrate using the trigonometric identities $\sin^2(x)=\frac12(1-\cos(2x))$, $\cos^2(x)=\frac12(1+\cos(2x))$, $1+\tan^2(x)=\sec^2(x)$ and $\cot^2(x)+1=\cosec^2(x)$.
- Establish and use the formula $\int \sec^2(x)\,dx=\tan(x)+c$.
- Use substitution $u=g(x)$ to integrate expressions of the form $f(g(x))g'(x)$.
- Establish and use the formula $\int \frac1x\,dx=\ln|x|+c$ for $x\ne0$ and $\int \frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c$ for $f(x)\ne0$.
Specialist integration starts by widening the set of patterns you can recognise. The aim is not to memorise a giant list. It is to notice when an integrand can be reshaped into a known derivative, identity or substitution.
Original Sylligence diagram for specialist integration techniques.
Trigonometric square identities
The identities
$ \sin^2x=\frac12(1-\cos2x) $
and
$ \cos^2x=\frac12(1+\cos2x) $
turn squared trigonometric functions into expressions you can integrate directly.
For example:
$ \int \cos^2x\,dx =\frac12\int(1+\cos2x)\,dx $
$ =\frac{x}{2}+\frac{\sin2x}{4}+c. $
The same idea works for $\sin^2(ax+b)$ or $\cos^2(ax+b)$, but remember the chain-rule factor.
The reciprocal trigonometric identities also matter:
$ 1+\tan^2x=\sec^2x $
and
$ \cot^2x+1=\cosec^2x. $
They are especially useful when an integrand can be rewritten as the derivative of $\tan x$ or $-\cot x$. QCAA explicitly includes:
$ \int \sec^2x\,dx=\tan x+c. $
You should also recognise the companion logarithmic integral:
$ \int \tan x\,dx=-\ln|\cos x|+c=\ln|\sec x|+c. $
That result comes from writing $\tan x=\frac{\sin x}{\cos x}$ and using $u=\cos x$. It is a good reminder that trigonometric and logarithmic methods are not separate boxes. A trig integral can become a log integral once the right substitution is visible.
Powers of sine and cosine
Integrals such as $\int \sin^m x\cos^n x\,dx$ are mostly about parity. The powers tell you which identity or substitution to use.
If the power of sine is odd, save one $\sin x$ factor for $du$ and convert the remaining even power of sine using:
$ \sin^2x=1-\cos^2x. $
Then use $u=\cos x$. For example:
$ \int \sin^3x\cos^2x\,dx =\int (1-\cos^2x)\cos^2x\sin x\,dx. $
With $u=\cos x$, $du=-\sin x\,dx$, so the integral becomes:
$ -\int (1-u^2)u^2\,du. $
If the power of cosine is odd, save one $\cos x$ factor and convert the remaining even power of cosine using:
$ \cos^2x=1-\sin^2x. $
Then use $u=\sin x$.
If both powers are odd, either approach can work. Save one of the odd factors, convert the rest, and choose the substitution that makes the algebra cleaner. If both powers are even, use the power-reduction identities:
$ \sin^2x=\frac12(1-\cos2x),\quad \cos^2x=\frac12(1+\cos2x). $
For $\tan^m x\sec^n x$ integrals, a common Specialist move is to save a $\sec^2x$ factor when the power of secant is even. Then convert the remaining secant factors into tangents using:
$ \sec^2x=1+\tan^2x $
and let $u=\tan x$. For example, $\int \tan^3x\sec^4x\,dx$ can be written as:
$ \int \tan^3x\sec^2x(1+\tan^2x)\,dx, $
then $u=\tan x$ turns it into a polynomial integral.
Substitution
Substitution is the reverse of the chain rule. It is useful when you see a function and a version of its derivative nearby.
For example:
$ \int x\sqrt{9-x^2}\,dx. $
Let
$ u=9-x^2. $
Then
$ du=-2x\,dx $
so
$ x\,dx=-\frac12du. $
Therefore:
$ \int x\sqrt{9-x^2}\,dx =-\frac12\int u^{1/2}\,du =-\frac13u^{3/2}+c. $
Substitute back:
$ -\frac13(9-x^2)^{3/2}+c. $
For definite integrals, you have two valid options. You can change the limits into $u$-values and stay in the new variable, or integrate in $u$ and then substitute back into $x$ before using the original limits. Do not mix those two approaches in one line.
The same idea handles integrals where the derivative is not sitting there perfectly. For example:
$ \int x^3\sqrt{9-x^2}\,dx. $
Let $u=9-x^2$. Then $du=-2x\,dx$ and $x^2=9-u$. Rewrite $x^3\,dx$ as $x^2(x\,dx)$:
$ \int x^3\sqrt{9-x^2}\,dx =-\frac12\int (9-u)u^{1/2}\,du. $
Now expand and integrate:
$ -\frac12\int (9u^{1/2}-u^{3/2})\,du =-3u^{3/2}+\frac15u^{5/2}+c. $
Substituting back gives:
$ -3(9-x^2)^{3/2}+\frac15(9-x^2)^{5/2}+c. $
For definite substitution, change every part of the integral. If $u=9-x^2$ and $x$ runs from $0$ to $2$, then $u$ runs from $9$ to $5$. The new integral uses $u$-limits and no remaining $x$ terms. This avoids the common half-substitution error where the integrand is in $u$ but the limits are still in $x$.
Linear substitutions also need care. In an expression such as $\int \frac{x}{\sqrt{2x-1}}\,dx$, using $u=2x-1$ gives $x=\frac{u+1}{2}$ and $dx=\frac12du$. The numerator cannot just be left as $x$.
Logarithmic integrals
Specialist uses the full logarithmic integral:
$ \int \frac1x\,dx=\ln|x|+c. $
The absolute value matters because the derivative of $\ln|x|$ is $\frac1x$ for $x\ne0$.
A very common pattern is:
$ \int \frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c. $
Choosing between the patterns
Start by checking whether the integrand simplifies. A squared sine or cosine often wants a double-angle identity. A composite function with its derivative nearby wants substitution. A quotient whose numerator is the derivative of the denominator wants a logarithm.
If the numerator is almost but not exactly the derivative of the denominator, split it. For example:
$ \int \frac{4x+5}{x^2+2x+3}\,dx $
has denominator derivative $2x+2$. Rewrite the numerator as a multiple of $2x+2$ plus a leftover constant, then integrate the logarithmic part first.