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QLDSpecialist MathematicsQuick questions

Unit 4: Further calculus, and statistical inference

Quick questions on Integration techniques: substitution and partial fractions (QCE Specialist Mathematics Unit 4)

6short 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 integration by substitution?
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Substitution reverses the chain rule. If the integrand contains a function and (a multiple of) its derivative, let uu be the inner function. With u=g(x)u = g(x) and du=g′(x) dxdu = g'(x)\,dx,
What are integration by partial fractions?
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A proper rational function with a factorisable denominator can be split into simpler fractions. For distinct linear factors,
What is volumes of solids of revolution?
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Rotating the region under y=f(x)y = f(x) between x=ax = a and x=bx = b about the xx-axis produces a solid of volume
What is choose the substitution?
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Let u=x2+1u = x^2 + 1, so dudx=2x\dfrac{du}{dx} = 2x, giving du=2x dxdu = 2x\,dx. The numerator 2x dx2x\,dx becomes dudu exactly.
What are change the limits?
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When x=0x = 0, u=02+1=1u = 0^2 + 1 = 1. When x=1x = 1, u=12+1=2u = 1^2 + 1 = 2.
What is check by a second method?
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The integrand is ddxln⁡(x2+1)\dfrac{d}{dx}\ln(x^2+1), so the antiderivative is ln⁡(x2+1)\ln(x^2+1). Evaluating directly: ln⁡(2)−ln⁡(1)=ln⁡2\ln(2) - \ln(1) = \ln 2. The two methods agree, so the answer is ln⁡2≈0.693\ln 2 \approx 0.693.

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