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Further Calculus

Quick questions on Further integration techniques explained: H2 Further Mathematics

8short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is completing the square?
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When the quadratic is not yet in a2±(linear)2a^2 \pm (\text{linear})^2 form, complete the square first. For example 1x2+4x+13=1(x+2)2+9\dfrac{1}{x^2 + 4x + 13} = \dfrac{1}{(x + 2)^2 + 9}, which then matches the arctan form with a=3a = 3 and a shift x→x+2x \to x + 2.
What are trigonometric substitutions?
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The substitution is chosen to make the surd a single trig function via a Pythagorean identity:
What are hyperbolic substitutions?
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For a2+x2\sqrt{a^2 + x^2} and x2−a2\sqrt{x^2 - a^2}, hyperbolic substitutions x=asinh⁡ux = a\sinh u and x=acosh⁡ux = a\cosh u exploit cosh⁡2u−sinh⁡2u=1\cosh^2 u - \sinh^2 u = 1, often leading to a logarithmic form for the result. These give the inverse hyperbolic functions, which can be written as logarithms.
What are integrals leading to logarithms?
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A fraction whose numerator is the derivative of the denominator integrates to a logarithm: ∫f′(x)f(x) dx=ln⁡∣f(x)∣+C\displaystyle\int \frac{\mathrm{f}'(x)}{\mathrm{f}(x)}\,dx = \ln|\mathrm{f}(x)| + C. Spotting this pattern (or engineering it by adjusting a constant) avoids unnecessary substitution.
What is sign of the surd?
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a2−a2sin⁡2θ=acos⁡θ\sqrt{a^2 - a^2\sin^2\theta} = a\cos\theta only when cos⁡θ≥0\cos\theta \geq 0; keep θ\theta in the principal range.
What is q1?
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Write ∫19−x2 dx\displaystyle\int \frac{1}{\sqrt{9 - x^2}}\,dx. [1 mark]
What is q2?
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Which substitution suits ∫1+x2 dx\displaystyle\int \sqrt{1 + x^2}\,dx? [1 mark]
What is q3?
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Complete the square for the denominator of ∫1x2−2x+5 dx\displaystyle\int \frac{1}{x^2 - 2x + 5}\,dx. [1 mark]

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