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QLDSpecialist MathematicsQuick questions

Unit 4: Further calculus, and statistical inference

Quick questions on Integration by parts and trigonometric integrals in QCE Specialist Mathematics Unit 4

5short 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 are integration by parts?
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Integration by parts reverses the product rule. For functions uu and vv,
What is choosing uu with LIATE?
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The order Logarithmic, Inverse-trig, Algebraic, Trigonometric, Exponential (LIATE) is a reliable guide: pick uu to be whichever factor appears first in that list, since differentiating it tends to simplify. For ∫xcos⁡x dx\int x\cos x\, dx, the algebraic xx comes before the trigonometric cos⁡x\cos x, so u=xu = x and dv=cos⁡x dxdv = \cos x\, dx.
What are repeated parts?
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Sometimes you apply parts twice. For ∫x2ex dx\int x^2 e^x\, dx, the first application leaves ∫2xex dx\int 2x e^x\, dx, which needs parts again. Each pass lowers the power of the algebraic factor until it disappears.
What is choose uu and dvdv by LIATE?
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Algebraic before trigonometric, so
What is check by differentiating?
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ddx(xsin⁡x+cos⁡x)=sin⁡x+xcos⁡x−sin⁡x=xcos⁡x.\dfrac{d}{dx}\big(x\sin x + \cos x\big) = \sin x + x\cos x - \sin x = x\cos x. This matches the integrand, confirming the result.

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