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MH HSC 12 Maths

Indefinite Integration

Each concept is a short lesson, then questions until you get three right.

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11 concepts, a short lesson each

  • Antiderivatives0/2
  • Substitution0/3
  • Quadratic forms0/3
  • Parts and partial fractions0/3

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Integration as the reverse of differentiation

∫f(x) dx=F(x)+c\int f(x)\,dx = F(x) + c when F′(x)=f(x)F'(x) = f(x); use the standard formulae term by term, e.g. ∫xn dx=xn+1n+1+c\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + c for n≠−1n \ne -1

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Antiderivatives0 of 2

  1. Integration as the reverse of differentiation∫f(x) dx=F(x)+c\int f(x)\,dx = F(x) + c when F′(x)=f(x)F'(x) = f(x); use the standard formulae term by term, e.g. ∫xn dx=xn+1n+1+c\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + c for n≠−1n \ne -1Start
  2. Rewriting before integratingExpand, divide out or use an identity first (sin⁡2x=1−cos⁡2x2\sin^2 x = \frac{1 - \cos 2x}{2}, tan⁡2x=sec⁡2x−1\tan^2 x = \sec^2 x - 1) so every term matches a standard formStart

Substitution0 of 3

  1. Integration by substitutionPut u=g(x)u = g(x) so that g′(x) dx=dug'(x)\,dx = du: ∫f(g(x)) g′(x) dx=∫f(u) du\int f(g(x))\,g'(x)\,dx = \int f(u)\,du; in particular ∫f′(x)f(x) dx=log⁡∣f(x)∣+c\int \dfrac{f'(x)}{f(x)}\,dx = \log|f(x)| + cStart
  2. Powers and products of trigonometric functionsTurn products into sums (2cos⁡Acos⁡B=cos⁡(A+B)+cos⁡(A−B)2\cos A\cos B = \cos(A + B) + \cos(A - B)); for sin⁡mxcos⁡nx\sin^m x\cos^n x with an odd power, keep one factor and substitute for the other functionStart
  3. The substitution t = tan(x/2)For ∫dxa+bsin⁡x\int \dfrac{dx}{a + b\sin x} or dxa+bcos⁡x\dfrac{dx}{a + b\cos x}, put t=tan⁡x2t = \tan\dfrac{x}{2}: sin⁡x=2t1+t2\sin x = \dfrac{2t}{1 + t^2}, cos⁡x=1−t21+t2\cos x = \dfrac{1 - t^2}{1 + t^2}, dx=2 dt1+t2dx = \dfrac{2\,dt}{1 + t^2}Start

Quadratic forms0 of 3

  1. Standard integrals of quadratic forms∫dxx2+a2=1atan⁡−1xa\int \dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\tan^{-1}\dfrac{x}{a}, ∫dxx2−a2=12alog⁡∣x−ax+a∣\int \dfrac{dx}{x^2 - a^2} = \dfrac{1}{2a}\log\left|\dfrac{x - a}{x + a}\right|, ∫dxa2−x2=sin⁡−1xa\int \dfrac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\dfrac{x}{a}, and the rest of the tableStart
  2. Completing the squareWrite ax2+bx+cax^2 + bx + c as a[(x+p)2±q2]a\left[(x + p)^2 \pm q^2\right], then use the standard form with x+px + p in place of xxStart
  3. Integrals of (px + q) over a quadraticWrite px+q=A ddx(ax2+bx+c)+Bpx + q = A\,\dfrac{d}{dx}(ax^2 + bx + c) + B; the AA part gives a log or a root, the BB part a standard quadratic formStart

Parts and partial fractions0 of 3

  1. Integration by parts∫u v dx=u∫v dx−∫(u′∫v dx)dx\int u\,v\,dx = u\int v\,dx - \int \left(u'\int v\,dx\right)dx; choose uu by LIATE (log, inverse trig, algebraic, trig, exponential)Start
  2. Special forms from integration by parts∫ex[f(x)+f′(x)]dx=exf(x)+c\int e^x\left[f(x) + f'(x)\right]dx = e^x f(x) + c; ∫a2−x2 dx\int \sqrt{a^2 - x^2}\,dx and its relatives; integrals that return to themselves, like ∫exsin⁡x dx\int e^x\sin x\,dxStart
  3. Integration by partial fractionsSplit a proper rational function into Ax−a+Bx−b+…\dfrac{A}{x - a} + \dfrac{B}{x - b} + \dots (repeated and quadratic factors too), then integrate each pieceStart

Practise Indefinite Integration

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions