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NSWMaths Extension 1Quick questions

Calculus (ME-C1, C2, C3)

Quick questions on Integrals giving inverse trig functions: arcsin⁡\arcsin, arctan⁡\arctan and the patterns to recognise

14short 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 standard antiderivatives?
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$∫11−x2 dx=arcsin⁡x+C,\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin x + C,$
What is generalising to a constant aa?
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$∫1a2−x2 dx=arcsin⁡xa+C(a>0),\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin \frac{x}{a} + C \quad (a > 0),$
What is recognising the patterns?
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The structure constant−x2\sqrt{\text{constant} - x^2} in the denominator with nothing else fancy in the numerator points to arcsin⁡\arcsin.
What is completing the square?
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If the denominator is 4+2x−x2\sqrt{4 + 2 x - x^2} or x2+4x+13x^2 + 4 x + 13, complete the square first to fit a standard form.
What is linear substitution shortcut?
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∫1a2−(bx+c)2 dx=1barcsin⁡bx+ca+C\int \frac{1}{\sqrt{a^2 - (b x + c)^2}} \, dx = \frac{1}{b} \arcsin \frac{b x + c}{a} + C.
What is coefficient inside square root?
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Evaluate ∫14−9x2 dx\int \frac{1}{\sqrt{4 - 9 x^2}} \, dx.
What is complete the square?
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Evaluate ∫1x2−6x+25 dx\int \frac{1}{x^2 - 6 x + 25} \, dx.
What is definite arctan?
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Evaluate ∫011x2+9 dx\int_0^1 \frac{1}{x^2 + 9} \, dx.
What is mixed with substitution?
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Evaluate ∫2x+1x2+x+2 dx\int \frac{2 x + 1}{x^2 + x + 2} \, dx.
What is when the numerator is not a derivative?
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Evaluate ∫1x2+x+1 dx\int \frac{1}{x^2 + x + 1} \, dx.
What is sign confusion with arccos⁡\arccos?
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∫−11−x2 dx=arccos⁡x+C\int \frac{-1}{\sqrt{1 - x^2}} \, dx = \arccos x + C. Most people prefer to keep arcsin⁡\arcsin and handle the sign outside.
What is domain issues?
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11−x2\frac{1}{\sqrt{1 - x^2}} is undefined for ∣x∣>1|x| > 1. A definite integral with limits outside [−1,1][-1, 1] is invalid.
What is confusing a2+x2a^2 + x^2 with a2−x2a^2 - x^2?
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Plus gives arctan⁡\arctan, minus gives arcsin⁡\arcsin (with the square root). Easy to mix up under pressure.
What is forgetting to substitute for dxdx?
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If the substitution introduces a constant scaling, that constant ends up dividing the antiderivative. :::

All Maths Extension 1Q&A pages