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NSWMaths Extension 1Quick questions

Calculus (ME-C1, C2, C3)

Quick questions on Integration by substitution in HSC Maths Extension 1: choosing uu, transforming the integral and changing limits

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 the general method?
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For an integral ∫f(g(x))g′(x) dx\int f(g(x)) g'(x) \, dx:
What is choosing a good uu?
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Look for one of these patterns in the integrand:
What is changing limits?
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For ∫abf(g(x))g′(x) dx\int_a^b f(g(x)) g'(x) \, dx, when u=g(x)u = g(x), the new limits are u=g(a)u = g(a) at the bottom and u=g(b)u = g(b) at the top.
What is linear inside argument?
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For ∫f(ax+b) dx\int f(a x + b) \, dx, the shortcut is ∫f(ax+b) dx=1aF(ax+b)+C\int f(a x + b) \, dx = \frac{1}{a} F(a x + b) + C, where FF is the antiderivative of ff.
What is reverse chain rule patterns?
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Memorise these common patterns:
What is logarithm pattern?
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Evaluate ∫2xx2+1 dx\int \frac{2 x}{x^2 + 1} \, dx.
What is trig inside?
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Evaluate ∫sin⁡2xcos⁡x dx\int \sin^2 x \cos x \, dx.
What is definite integral with limit change?
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Evaluate ∫02xex2 dx\int_0^2 x e^{x^2} \, dx.
What is mixed substitution?
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Evaluate ∫x1−x dx\int x \sqrt{1 - x} \, dx.
What is linear inside argument shortcut?
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Evaluate ∫e2x+1 dx\int e^{2 x + 1} \, dx.
What is forgetting to change the limits?
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Either change limits to uu-values, or substitute back to xx before evaluating. Do not mix the two; substituting uu-values into the back-substituted xx expression gives wrong answers.
What is picking the wrong uu?
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If your chosen uu does not eliminate xx from the integral, the substitution failed. Try u=u = a different inner function.
What is dropping a constant when du=k dxdu = k \, dx?
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du=2x dxdu = 2 x \, dx means x dx=12dux \, dx = \frac{1}{2} du, so the integral picks up a 12\frac{1}{2}.
What is missing the absolute value in ln⁡\ln?
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∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln |x| + C in general; you can drop the absolute value only when x>0x > 0 or x<0x < 0 throughout the interval of integration. :::

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