You are given that $I=\displaystyle\int{x{{\left( 1-x \right)}^{\frac{1}{2}}}}\mathrm{d}x$. Use inte

Question

You are given that $I=\displaystyle\int{x{{\left( 1-x \right)}^{\frac{1}{2}}}}\mathrm{d}x$. Use integration by parts to find an expression for $I$. [2] Use the substitution ${{u}^{2}}=1-x$ to find another expression for $I$. [2] Show algebraically that your answers to parts (i) and (ii) differ by a constant. [2]

Accepted Answer

(i) -2x(1-x)^(3/2)/3 - 4(1-x)^(5/2)/15 + C₁; (ii) -2(1-x)^(3/2)/3 + 2(1-x)^(5/2)/5 + C₂; (iii) Difference = constant D (shown equivalent)

N2019 P2 Q1

N2019 P2 Q1

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