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Simplify e^(k log f(x)) to f(x)^k before integrating

This type of question, asked 3 times: 19 Apr 2025 S1, 20 Apr 2026 S1, 12 May 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    19 Apr 2025 S1

    112
    ∫e2030log⁡x−e2029log⁡xe2028log⁡x−e2027log⁡xdx=…\int\frac{e^{2030\log x}-e^{2029\log x}}{e^{2028\log x}-e^{2027\log x}}dx = \ldots.

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  2. Question 2

    20 Apr 2026 S1

    79
    ∫(elog⁡(sin⁡x)+cos⁡x)x dx=\displaystyle\int\left(e^{\log(\sin x)} + \cos x\right)x\,dx =

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  3. Question 3

    12 May 2026 S1

    80
    If ∫sin⁡4x⋅elog⁡(cos⁡x) dx=1ksin⁡kx+C\displaystyle\int\sin^4 x \cdot e^{\log(\cos x)}\,dx = \dfrac{1}{k}\sin^k x + C, for x∈(−π2,π2)x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right), then ...

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