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LearnAQA7357First teaching 2017Paper 1 · PureIntegration

Paper 1 · Module

Integration

13 subtopics≈ 545 min in total

Start the module: Integrating eˣ, 1/x and trigonometric functions Open graphing calculator

Integrating eˣ, 1/x and trigonometric functions
Reversing the new derivatives: $\int e^{ax+b}\,dx$ , $\int\dfrac{1}{ax+b}\,dx$ (the missing case $n=-1$ ), and $\int\sin$ , $\cos$ and $\sec^2$ , always remembering to divide by $a$ .
≈ 40 min
Indefinite integration
Integration as differentiation in reverse: integrating powers of $x$ , why there is always a $+c$ , and how one point pins down the exact curve.
≈ 40 min
Integration using trigonometric identities
Some trigonometric expressions can't be integrated as they stand. Rewriting them with identities, such as $\cos^2x=\tfrac12(1+\cos2x)$ and $\tan^2x=\sec^2x-1$ , turns them into standard integrals.
≈ 40 min
Integrating (ax + b)ⁿ
How to integrate a power of a linear bracket in one step by reversing the chain rule: raise the power, divide by the new power, and divide by the coefficient of $x$ .
≈ 35 min
Exact definite integrals and logarithms
Evaluating definite integrals exactly when the answer involves $e$ , $\ln$ or $\pi$ ; simplifying logarithmic answers; spotting $\dfrac{f'(x)}{f(x)}$ ; and integrating with partial fractions.
≈ 45 min
Definite integrals
What $\displaystyle\int_a^b f(x)\,dx$ means and how to evaluate it, why there is no $+c$ , how to find an unknown limit, and how some integrals with an infinite limit still give a finite answer.
≈ 40 min
Areas under and between curves
Using definite integrals to find areas: under a curve, regions below the axis, between a curve and a line, between two curves, and against the $y$ -axis.
≈ 50 min
Integration by substitution
The reverse of the chain rule: replace part of the integrand by $u$ , change $dx$ into $du$ , change the limits, and turn a hard integral into an easy one.
≈ 50 min
Integration by parts
The reverse of the product rule: $\displaystyle\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$ . How to choose $u$ , when to use it twice, and the trick for $\int\ln x\,dx$ .
≈ 50 min
Forming differential equations
Turn words about change into mathematics. "Grows in proportion to", "decreases at a rate inversely proportional to", "flows in and leaks out": each becomes an equation for a derivative, ready to solve.
≈ 30 min
Separating the variables
Gather every $y$ with $dy$ on one side and every $x$ with $dx$ on the other, integrate, and add one constant. The method that solves almost every Paper 3 differential equation.
≈ 35 min
Particular solutions with harder integrals
Exam differential equations are integration questions in disguise. Once the variables are separated, one side often needs partial fractions, integration by parts or a trigonometric identity. Bring all of Paper 3 integration to bear, then pin down the constant.
≈ 45 min
Modelling with differential equations
Populations, cooling drinks, draining tanks and spreading rumours. Solve the equation, use two pieces of data to fix the constants, and then ask the real questions: when, how much, and what happens in the long run?
≈ 45 min