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Integral with fractional powers of x, substituting x = t^n, given f(0): find f(1)

This type of question, asked 2 times: 28 Jan 2025, 28 Jan 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    28 Jan 2025

    136
    If f(x)=∫1x1/4(1+x1/4)dx,f(0)=−6f(x) = \int\frac{1}{x^{1/4}\left( 1 +x^{1/4} \right)}dx,f(0) = - 6, then f(1)f(1) is equal to :

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  2. Question 2

    28 Jan 2026 S2

    69
    Let f(x)=∫ dxx(23)+2x(12)f(x) =\int_{}^{}\ \frac{dx}{x^{\left( \frac{2}{3} \right)}+ 2x^{\left( \frac{1}{2} \right)}} be such that f(0)=−26+24log⁡e(2)f(0) = - 26 + 24\log_{e}(2). If f(1)=a+blog⁡e(3)f(1) =a+b\log_{e}(3), where a,b∈Za,b\in Z, then a+ba + b is equal to:

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