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MH HSC 12 Maths

Definite Integration

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

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7 concepts, a short lesson each

  • Evaluating0/4
  • Properties0/3

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Definite integral as the limit of a sum

∫abf(x) dx=lim⁡n→∞∑r=1nh f(a+rh)\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{r=1}^{n} h\,f(a + rh) with h=b−anh = \dfrac{b - a}{n}: the total area of thin rectangles

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Evaluating0 of 4

  1. Definite integral as the limit of a sum∫abf(x) dx=lim⁡n→∞∑r=1nh f(a+rh)\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{r=1}^{n} h\,f(a + rh) with h=b−anh = \dfrac{b - a}{n}: the total area of thin rectanglesStart
  2. The fundamental theorem of integral calculusIf F′(x)=f(x)F'(x) = f(x), then ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a): find any antiderivative and subtract its values at the limitsStart
  3. Substitution in a definite integralWhen you put u=g(x)u = g(x), change the limits too: x=ax = a becomes u=g(a)u = g(a), x=bx = b becomes u=g(b)u = g(b); then never go back to xxStart
  4. Parts and partial fractions with limits∫abu v dx=[u∫v dx]ab−∫ab(u′∫v dx)dx\int_a^b u\,v\,dx = \left[u\int v\,dx\right]_a^b - \int_a^b \left(u'\int v\,dx\right)dx; split rational functions first, then evaluateStart

Properties0 of 3

  1. Basic properties and splitting the interval∫ab=−∫ba\int_a^b = -\int_b^a, ∫ab=∫ac+∫cb\int_a^b = \int_a^c + \int_c^b; split at the point where ∣x−c∣|x - c| or a piecewise function changes formStart
  2. The property f(a + b − x)∫abf(x) dx=∫abf(a+b−x) dx\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx; add the two forms when they sum to something simple, as in ∫0π/2sin⁡xsin⁡x+cos⁡x dx=π4\int_0^{\pi/2}\dfrac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx = \dfrac{\pi}{4}Start
  3. Odd and even functions, and the 2a property∫−aaf(x) dx\int_{-a}^{a} f(x)\,dx is 00 for odd ff and 2∫0af(x) dx2\int_0^a f(x)\,dx for even ff; ∫02af(x) dx=2∫0af(x) dx\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx when f(2a−x)=f(x)f(2a - x) = f(x)Start

Practise Definite Integration

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

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