Game plan 6 · Unit 4
Integration
One flowchart decides the technique. Everything else on this page is the technique itself.
Reach for it when
You see any integral sign, “area between”, “volume of revolution”, “Simpson’s rule”, or an exponential probability density function.
Part 1
The integration flowchart
Fractions are where most integration marks are lost, so quotients get their own steps in Part 2.
Already a Methods form or a formula-sheet form. Integrate it.
Even powers of sine or cosine, or a product of trig functions. Rewrite with an identity (half-angle for $\sin^2$ and $\cos^2$, or a product identity).
Run the three checks of Game plan 6.1 in order. A formula-sheet form gives $\sin^{-1}$ or $\tan^{-1}$, and a top that is the bottom’s derivative gives $k\ln|\text{bottom}|+c$. Both finish it. Two distinct linear factors mean partial fractions, then loop.
Is one factor the derivative of something inside the other? If so, substitute, then loop. If not, use integration by parts (Game plan 6.3), choosing $u$ with DETAIL.
Rearrange it into a form you know. Split the fraction, or long-divide an improper fraction, then loop.
Work down the list and stop at the first step that matches. When a step transforms the integral, start again at step 1 with the new integral.
Memorise
$\sin^2x=\frac12(1-\cos2x)$ and $\cos^2x=\frac12(1+\cos2x)$ are not on the sheet. It only gives the double-angle forms, so if you can’t remember them, rearrange $\cos2A=1-2\sin^2A$. Every even trig power in the course goes through these two lines.
In the Topic Guides
Part 2
Quotients
Quotients: three checks, in order
- 1Is it a formula-sheet form? $\frac{1}{\sqrt{a^2-x^2}}$ gives $\sin^{-1}\frac xa$ and $\frac{a}{a^2+x^2}$ gives $\tan^{-1}\frac xa$, straight off the sheet. Watch the value of $a$: write $a^2=\dots$, so $a=\dots$.
- 2Is the top the bottom’s derivative, or a constant multiple of it? Then the answer is $k\ln|\text{bottom}|+c$.
- 3Are there two distinct linear factors underneath? Use partial fractions: set $\frac{f(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}$, solve for the constants, and loop. Each piece becomes a log.
- 4Is it an improper fraction, with the degree of the top at least the degree of the bottom? Long-divide first, then run the checks again.
In the Topic Guides
Part 3
Substitution and integration by parts
Most integration is spotting a derivative you already know and working backwards. If you can name the derivative that would produce the integrand, you are finished.
Substitution
- 1Look for the shape $f(g(x))\,g'(x)$: an inside function whose derivative sits, up to a constant, as a factor.
- 2Set $u=g(x)$, work out $du$, and substitute completely, with no stray $x$s.
- 3For a definite integral, convert the bounds to $u$-values and never go back to $x$.
- 4Integrate in $u$, then substitute back (indefinite integrals only).
Integration by parts
- 1Pick $u$ first, using DETAIL: Derivative of $u$, Exponential, Trig, Algebraic, Inverse trig, Logarithm. Work down the list and take the first one you can see in the integrand. Whatever is left over is $\frac{dv}{dx}$.
- 2Check the choice: $u$ should get simpler when you differentiate it. If it doesn’t, you have picked them the wrong way round.
- 3Apply the sheet formula $\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$.
- 4If the remaining integral still needs work, go back into the flowchart with it.
In the Topic Guides
Part 4
Trig powers
Integrating trig powers
- 1Even powers of sine or cosine: use the half-angle identities, then integrate term by term.
- 2Odd powers: split off one factor, convert the rest with $\sin^2+\cos^2=1$, and substitute.
- 3$\tan^2x$ becomes $\sec^2x-1$ (a Pythagorean identity on the sheet), because $\int\sec^2x\,dx=\tan x+c$ is on the sheet.
- 4Products such as $\sin A\cos B$ use the product identities on the sheet to turn them into sums.
In the Topic Guides
Part 5
Areas and volumes
Areas and volumes
- 1Sketch. Every area and volume mark starts with knowing which curve is on top and where the curves cross.
- 2Area between curves: find the intersection points by equating, then use $\int(\text{top}-\text{bottom})\,dx$.
- 3Rotation about the $x$-axis: make sure the function and the bounds are both in terms of $x$, then $V=\pi\int_a^b(f(x))^2\,dx$.
- 4Rotation about the $y$-axis: make sure the function and the bounds are both in terms of $y$. Substitute the original bounds into the rearranged equation to convert them, then $V=\pi\int_a^b(f(y))^2\,dy$.
- 5Rotating a region between two curves: define the region with the area steps first, then $V=\pi\int_a^b\left(\text{outer}^2-\text{inner}^2\right)dx$.
Part 6
Simpson’s rule
Using Simpson’s rule
- 1Check that $n$ is even, and work out $w=\frac{b-a}{n}$.
- 2Write out $x_0,\dots,x_n$ and their function values in a table before you touch the formula.
- 3Apply the sheet’s weights, $1,4,2,4,\dots,4,1$. The values at odd indices get the 4s.
The worded form, for when you are under pressure
The first and last values stand alone. Every value at an odd index is multiplied by 4, and every value at an even index by 2. The width is $w=\frac{b-a}{n}$, where $n$ is the number of strips.
Tech hint
On Paper 2, build the table with your calculator’s table-of-values function rather than working out each value by hand.
In the Topic Guides
Part 7
Exponential probability density functions
Waiting-time questions are integration questions set in a real context. The distribution is on the sheet, and almost all of the work is finding $\lambda$ and then integrating between the right two limits.
The distribution · On the sheet
The mean and standard deviation are equal. That is how you recognise this distribution, and if a question gives you either one, you have $\lambda$ in one line.
The exponential distribution
- 1The PDF $f(t)=\lambda e^{-\lambda t}$, the mean and the standard deviation are on the sheet.
- 2Find probabilities by integrating the PDF: $P(T\le t)=1-e^{-\lambda t}$ and $P(T\gt t)=e^{-\lambda t}$.
- 3For quantiles, set $P(T\le m)=q$ and solve with logs. The median is $\frac{\ln2}{\lambda}$.
- 4Finding $\lambda$ is the hidden work in most of these questions. Game plan 6.7a has the three ways in.
Find $\lambda$ first: three routes in
- 1The mean is given: “on average”, “a mean of”, “typically every”. Then $\lambda=\frac1\mu$.
- 2The standard deviation is given. It’s the same line, $\lambda=\frac1\sigma$, because $\mu=\sigma$ here.
- 3Neither is given, and a probability is hidden in the wording. This is the complex unfamiliar version: “30% last longer than 5 minutes”, “half are served within 2 minutes”. Turn the sentence into one equation, such as $P(T\gt5)=0.3$ or $P(T\le2)=0.5$, and solve for $\lambda$ with logs.
- 4Then fix the units. $\lambda$ is a rate per unit of $t$: a mean of 4 minutes gives $\lambda=0.25$ per minute, so every later time must also be in minutes. Convert before you integrate, never after.
1 · A probability between two times
Integrate the PDF between the two limits, or subtract tails: $P(a\le T\le b)=e^{-\lambda a}-e^{-\lambda b}$. The one-sided versions are $P(T\gt t)=e^{-\lambda t}$ and $P(T\le t)=1-e^{-\lambda t}$. If $\lambda$ isn’t given, run Game plan 6.7a first.
2 · A bound from a probability
The unknown is a limit, not a probability. Write the equation, then take logs: $1-e^{-\lambda t}=q\Rightarrow t=-\frac{\ln(1-q)}{\lambda}$. The median is the $q=0.5$ case: $t=\frac{\ln2}{\lambda}$.
3 · The mean or standard deviation
Both are $\frac1\lambda$. The Methods definitions $\int t\,f(t)\,dt$ and $\int t^2f(t)\,dt-\mu^2$ get there, but they are slow and easy to fumble under time. Quote $\frac1\lambda$ unless the question says “using the definition”.
4 · A worded problem mixing all three
The usual multi-part question. A sentence gives you $\lambda$, part a) wants a probability, part b) wants a time or a median, and part c) asks you to interpret in context. Answer c) in a sentence with units and the variable named. Bare numbers lose marks.
Watch for
“At least”, “more than” and “longer than” are the tail, $e^{-\lambda t}$, with no subtraction needed. Reaching for $1-e^{-\lambda t}$ out of habit is the most common error here.
Remember
The PDF is piecewise: $f(t)=0$ for $t\lt0$. A lower limit below zero is never $-\infty$. It is $0$.
Tech hint
On Paper 2, solve for $\lambda$ or a bound numerically and keep the unrounded value in memory for later parts. On Paper 1, the numbers are chosen so the logs simplify. If yours don’t, check which side of the curve you integrated.
Formulas
Formula panel
Each formula is labelled with how to treat it in the exam.
Watch for
The derivative of $\cos x$ and the integral of $\sin x$ both carry a negative sign. Check the sheet rather than assuming.
Tip
$1+\tan^2A=\sec^2A$ and $\cot^2A+1=\operatorname{cosec}^2A$ are under the Pythagorean identities on the sheet, not under calculus. Know where each formula is on the sheet.
Remember
Specialist writes $\int\frac1x\,dx=\ln|x|+c$. Keep the absolute value unless you know the inside is positive.
QCAA language
What the question says, and what it means
| Question type | What you’ll see | What it means |
|---|---|---|
| Choosing a technique | A bare integral | Run the flowchart |
| Quotient | Any fraction | Game plan 6.1. Three checks in order |
| “Hence” | “Hence, integrate …” | The previous part built your substitution or identity, so use it |
| Area or volume | “Region bounded by”, “rotated about” | Game plan 6.5. Sketch first |
| Approximation | “Use Simpson’s rule with $n=\dots$” | Game plan 6.6. Table first |
| Waiting times | $\lambda$, “exponential”, “time until” | Game plan 6.7. Find $\lambda$ first |
Spot a mistake on this page? Report it