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Polynomial times a power of a linear expression (or of 1 + x^2): substitute t for it and match coefficients

This type of question, asked 7 times: 2 May 2023 S2, 3 May 2023 S1, 10 May 2023 S1, 11 May 2023 S1, 14 May 2024 S1, 14 May 2024 S2, 15 May 2026 S2. All 7 questions below, oldest first.

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  1. Question 1

    2 May 2023 S2

    106
    If ∫x+12x−1 dx=f(x)2x−1+c\int \frac{x+1}{\sqrt{2x-1}}\,dx = f(x)\sqrt{2x-1} + c, then f(x)f(x) is equal to

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  2. Question 2

    3 May 2023 S1

    123
    If ∫(2x+4)x−1 dx=a(x−1)5/2+b(x−1)3/2+c\int(2x+4)\sqrt{x-1}\,dx=a(x-1)^{5/2}+b(x-1)^{3/2}+c where cc is a constant of integration, then the value of (2a+b)(2a+b) is

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  3. Question 3

    10 May 2023 S1

    124
    If ∫(2x+4)x−1 dx=a(x−1)5/2+b(x−1)3/2+c\int(2x+4)\sqrt{x-1}\,dx=a(x-1)^{5/2}+b(x-1)^{3/2}+c (where c is a constant of integration), then the value of a+ba+b is

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  4. Question 4

    11 May 2023 S1

    106
    If ∫(7x−2)3x+2 dx=A(3x+2)5/2+B(3x+2)3/2+c\int(7x-2)\sqrt{3x+2}\,dx = A(3x+2)^{5/2} + B(3x+2)^{3/2} + c (where c is a constant of integration), then the values of A and B are respectively

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  5. Question 5

    14 May 2024 S1

    106
    If ∫x3 dx1+x2=a(1+x2)1+x2+b1+x2+c\int\dfrac{x^3\,dx}{\sqrt{1+x^2}}=a(1+x^2)\sqrt{1+x^2}+b\sqrt{1+x^2}+c (Where c is a constant of integration), then the value of 3ab3ab is:

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  6. Question 6

    14 May 2024 S2

    145
    If ∫x21−x dx=p1−x(3x2+4x+8)+c\int x^2\sqrt{1-x}\,dx = p\sqrt{1-x}(3x^2+4x+8) + c, then the value of pp is

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  7. Question 7

    15 May 2026 S2

    81
    If ∫x(a−x)9 dx=[f(x)]1111−a[f(x)]1010+c\int x(a - x)^9\, dx = \dfrac{[f(x)]^{11}}{11} - a\dfrac{[f(x)]^{10}}{10} + c, where cc is a constant of integration, then f′(x)=⋯f'(x) = \cdots

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