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VICSpecialist MathematicsQuick questions

Unit 4: Calculus

Quick questions on Integration techniques: VCE Specialist Mathematics Unit 4

7short 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 choose the substitution?
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Let u=x2+1u = x^2 + 1. Then dudx=2x\dfrac{\mathrm{d}u}{\mathrm{d}x} = 2x, so x dx=12 dux\,\mathrm{d}x = \tfrac{1}{2}\,\mathrm{d}u. The integrand xx2+1 dxx\sqrt{x^2 + 1}\,\mathrm{d}x becomes u⋅12 du\sqrt{u}\cdot\tfrac12\,\mathrm{d}u.
What are change the terminals?
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When x=0x = 0, u=02+1=1u = 0^2 + 1 = 1. When x=2x = 2, u=22+1=5u = 2^2 + 1 = 5. So
What is integrate?
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12∫15u1/2 du=12⋅23[u3/2]15=13(53/2−13/2)\dfrac{1}{2}\int_1^5 u^{1/2}\,\mathrm{d}u = \dfrac{1}{2}\cdot\dfrac{2}{3}\Big[u^{3/2}\Big]_1^5 = \dfrac{1}{3}\Big(5^{3/2} - 1^{3/2}\Big).
What is evaluate?
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53/2=555^{3/2} = 5\sqrt{5} and 13/2=11^{3/2} = 1, so the value is 55−13\dfrac{5\sqrt{5} - 1}{3}.
What is q1?
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Evaluate ∫2x ex2 dx\displaystyle\int 2x\,e^{x^2}\,\mathrm{d}x. [2 marks]
What is q2?
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Evaluate ∫0111+x2 dx\displaystyle\int_0^{1} \frac{1}{1 + x^2}\,\mathrm{d}x. [2 marks]
What is q3?
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Evaluate ∫cos⁡2x dx\displaystyle\int \cos^2 x\,\mathrm{d}x. [2 marks]

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