Moderate -0.8 This is a straightforward algebraic manipulation followed by standard integration. Part (a) requires simple division of terms (splitting the fraction), and part (b) applies basic power rule integration. This is easier than average as it's a routine textbook exercise testing only fundamental skills with clear scaffolding.
3. (a) Express \(\frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } }\) in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants.
(b) Hence find
$$\int \frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } } d x$$
simplifying your answer.
Attempt to divide by \(2x^2\). Implied if either index or coefficient is correct
One correct term: \(\frac{1}{2}x\) or \(+2x^{-2}\)
A1
Allow \(\frac{1}{2}x^1 = 0.5x\) or \(+2x^{-2} = +\frac{2}{x^2}\)
\(\frac{1}{2}x + 2x^{-2}\) or \(0.5x + 2x^{-2}\)
A1
Accept \(x^1 = x\). Final answer of \(\frac{1}{2}x + \frac{2}{x^2}\) is M1A1A0
(b)
Answer
Marks
Guidance
Answer
Marks
Guidance
Raises any index by one for \(Ax^p + Bx^q\)
M1
One term correct and simplified: \(\frac{1}{4}x^2\) or \(0.25x^2\) or \(-2x^{-1}\) or \(-\frac{2}{x}\)
A1
\(\frac{1}{4}x^2 - 2x^{-1} + c\)
A1
Must include \(+c\). Accept \(0.25x^2 - \frac{2}{x} + c\) or \(\frac{x^3-8}{4x}+c\). Do not accept \(\frac{1}{4}x^2 + -2x^{-1}+c\)
## Question 3:
**(a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{x^3+4}{2x^2} = \frac{x^3}{2x^2}+\frac{4}{2x^2} = \frac{1}{2}x + 2x^{-2}$ | M1 | Attempt to divide by $2x^2$. Implied if either index or coefficient is correct |
| One correct term: $\frac{1}{2}x$ or $+2x^{-2}$ | A1 | Allow $\frac{1}{2}x^1 = 0.5x$ or $+2x^{-2} = +\frac{2}{x^2}$ |
| $\frac{1}{2}x + 2x^{-2}$ or $0.5x + 2x^{-2}$ | A1 | Accept $x^1 = x$. Final answer of $\frac{1}{2}x + \frac{2}{x^2}$ is M1A1A0 |
**(b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Raises any index by one for $Ax^p + Bx^q$ | M1 | |
| One term correct and simplified: $\frac{1}{4}x^2$ or $0.25x^2$ or $-2x^{-1}$ or $-\frac{2}{x}$ | A1 | |
| $\frac{1}{4}x^2 - 2x^{-1} + c$ | A1 | Must include $+c$. Accept $0.25x^2 - \frac{2}{x} + c$ or $\frac{x^3-8}{4x}+c$. Do not accept $\frac{1}{4}x^2 + -2x^{-1}+c$ |
---
3. (a) Express $\frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } }$ in the form $A x ^ { p } + B x ^ { q }$, where $A , B , p$ and $q$ are constants.\\
(b) Hence find
$$\int \frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } } d x$$
simplifying your answer.
\hfill \mbox{\textit{Edexcel C12 2017 Q3 [6]}}