The Weierstrass substitution

The Weierstrass substitution Quiz

Spec 2.5 · Edexcel A-Level Further Maths: Quiz

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5 questions, one answer each

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  1. 1. With t=tan⁡x2t=\tan\frac{x}{2}, dx\mathrm{d}x equals

    • A. dt1+t2\frac{\mathrm{d}t}{1+t^2}
    • B. 2 dt1−t2\frac{2\,\mathrm{d}t}{1-t^2}
    • C. 2 dt1+t2\frac{2\,\mathrm{d}t}{1+t^2}
    • D. 2(1+t2) dt2(1+t^2)\,\mathrm{d}t

  2. 2. ∫cosec⁡x dx\displaystyle\int\operatorname{cosec}x\,\mathrm{d}x equals

    • A. ln⁡∣tan⁡x2∣+c\ln\left|\tan\frac{x}{2}\right|+c
    • B. −cot⁡x+c-\cot x+c
    • C. ln⁡∣sin⁡x∣+c\ln|\sin x|+c
    • D. tan⁡x2+c\tan\frac{x}{2}+c

  3. 3. When x=π2x=\frac{\pi}{2}, the substitution gives t=t=

    • A. π4\frac{\pi}{4}
    • B. ∞\infty
    • C. 00
    • D. 11

  4. 4. ∫0π/211+cos⁡x dx\displaystyle\int_0^{\pi/2}\frac{1}{1+\cos x}\,\mathrm{d}x equals

    • A. 12\frac12
    • B. 11
    • C. 22
    • D. π2\frac{\pi}{2}

  5. 5. After the substitution, ∫11+sin⁡x−cos⁡x dx\displaystyle\int\frac{1}{1+\sin x-\cos x}\,\mathrm{d}x becomes

    • A. ∫11+t2 dt\displaystyle\int\frac{1}{1+t^2}\,\mathrm{d}t
    • B. ∫1t(t+1) dt\displaystyle\int\frac{1}{t(t+1)}\,\mathrm{d}t
    • C. ∫2t+1 dt\displaystyle\int\frac{2}{t+1}\,\mathrm{d}t
    • D. ∫1t dt\displaystyle\int\frac{1}{t}\,\mathrm{d}t

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