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Integral with a sum or difference of square roots in the denominator: rationalise

This type of question, asked 3 times: 31 Jan 2023, 27 Jan 2024, 2 Apr 2025. All 3 questions below, oldest first.

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  1. Question 1

    31 Jan 2023

    153
    Let α>0\alpha > 0. If ∫0αxx+α−xdx=16+20215\int_{0}^{\alpha} \frac{x}{\sqrt{x + \alpha}-\sqrt{x}}dx =\frac{16 + 20\sqrt{2}}{15}, then α\alpha is equal to:

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  2. Question 2

    27 Jan 2024

    69
    If ∫0113+x+1+xdx=a+b2+c3\int_{0}^{1} \frac{1}{\sqrt{3 + x}+\sqrt{1 + x}}dx = a + b\sqrt{2}+ c\sqrt{3}, where a,b,ca,b,c are rational numbers, then 2a+3b−4c2a + 3b - 4c is equal to :

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  3. Question 3

    2 Apr 2025

    138
    4∫01(13+x2+1+x2)dx−3log⁡e(3)4\int_{0}^{1} \left( \frac{1}{\sqrt{3 +x^{2}}+\sqrt{1 +x^{2}}} \right)dx - 3\log_{e}(\sqrt{3}) is equal to:

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