Moderate -0.8 This is a straightforward algebraic manipulation followed by routine integration using the power rule. Part (a) requires splitting the fraction and simplifying indices (standard C1 skill), and part (b) applies basic integration formulas with no problem-solving insight needed. Easier than average A-level questions.
6. (a) Show that \(\frac { x ^ { 2 } - 4 } { 2 \sqrt { } x }\) can be written in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants to be determined.
(b) Hence find
$$\int \frac { x ^ { 2 } - 4 } { 2 \sqrt { x } } \mathrm {~d} x , \quad x > 0$$
giving your answer in its simplest form.
M1: Divide by \(2\sqrt{x}\) to get exactly two terms. A1: Two of \(A,B,p,q\) correct from \(\frac{x^{\frac{3}{2}}}{2} - 2x^{-\frac{1}{2}}\). A1: Completely correct expression; powers must be simplified.
M1: Increases a fractional index by one. Do not allow if candidate integrates numerator and denominator of original separately. A1ft: One fractional term correct unsimplified (follow through on fractional index terms). A1: Both terms correct unsimplified.
## Question 6(a):
$\frac{x^2-4}{2\sqrt{x}} = \frac{x^2}{2\sqrt{x}} - \frac{4}{2\sqrt{x}} = \frac{1}{2}x^{\frac{3}{2}} - 2x^{-\frac{1}{2}}$ | M1 A1 A1 | M1: Divide by $2\sqrt{x}$ to get exactly two terms. A1: Two of $A,B,p,q$ correct from $\frac{x^{\frac{3}{2}}}{2} - 2x^{-\frac{1}{2}}$. A1: Completely correct expression; powers must be simplified.
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## Question 6(b):
$\int \frac{x^{\frac{3}{2}}}{2} - 2x^{-\frac{1}{2}}\,dx = \frac{x^{\frac{5}{2}}}{2\times2.5} - 2\frac{x^{\frac{1}{2}}}{0.5}(+c)$ | M1 A1ft A1 | M1: Increases a fractional index by one. Do not allow if candidate integrates numerator and denominator of original separately. A1ft: One fractional term correct unsimplified (follow through on fractional index terms). A1: Both terms correct unsimplified.
$= \frac{x^{\frac{5}{2}}}{5} - 4x^{\frac{1}{2}} + c$ | B1 | Fully simplified correct answer. Accept $0.2x^{2.5} - 4x^{0.5} + c$.
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6. (a) Show that $\frac { x ^ { 2 } - 4 } { 2 \sqrt { } x }$ can be written in the form $A x ^ { p } + B x ^ { q }$, where $A , B , p$ and $q$ are constants to be determined.\\
(b) Hence find
$$\int \frac { x ^ { 2 } - 4 } { 2 \sqrt { x } } \mathrm {~d} x , \quad x > 0$$
giving your answer in its simplest form.\\
\hfill \mbox{\textit{Edexcel C12 2016 Q6 [7]}}