Moderate -0.8 This is a straightforward integration question requiring basic techniques: simplifying algebraic fractions before integrating in part (i), and evaluating a definite integral with a parameter in part (ii). Both parts involve routine application of standard integration rules with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation required.
7. (i) Find
$$\int \frac { 2 + 4 x ^ { 3 } } { x ^ { 2 } } \mathrm {~d} x$$
giving each term in its simplest form.
(ii) Given that \(k\) is a constant and
$$\int _ { 2 } ^ { 4 } \left( \frac { 4 } { \sqrt { x } } + k \right) \mathrm { d } x = 30$$
find the exact value of \(k\).
Attempts to split the fraction. Can be awarded for \(\frac{2}{x^2}\) or \(\frac{4x^3}{x^2}\) or may be implied by the sight of one correct index e.g. \(px^{-2}\) or \(qx\) providing one of these terms is obtained correctly.
dM1: \(x^n \to x^{n+1}\) on any term. Dependent on the first M. A1: At least one term correct, simplified or un-simplified. Allow powers and coefficients to be un-simplified.
\(= -\frac{2}{x} + 2x^2 + c\)
A1
All correct and simplified including the \(+ c\). Accept equivalents such as \(-2x^{-1} + 2x^2 + c\)
M1: Integrates to obtain either \(\alpha x^{0.5}\) or \(kx\). A1: Correct integration (simplified or un-simplified). Allow powers and coefficients to be un-simplified e.g. \(4\frac{x^{-0.5+1}}{0.5}\). There is no need for \(+c\)
Substitutes both \(x=4\) and \(x=2\) into changed expression involving \(k\), subtracts either way round and sets equal to 30. Condone poor use or omission of brackets when subtracting.
ddM1: Attempts to solve for \(k\) from a linear equation in \(k\). Dependent upon both M's and need to have seen \(\int k\,dx \to kx\). A1: \(7+4\sqrt{2}\) or exact equivalent e.g. \(7+2^{2.5}\), \(7+4\times 2^{0.5}\)
# Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{2+4x^3}{x^2} = \frac{2}{x^2} + 4x = 2x^{-2} + 4x$ | M1 | Attempts to split the fraction. Can be awarded for $\frac{2}{x^2}$ or $\frac{4x^3}{x^2}$ or **may** be implied by the sight of one correct index e.g. $px^{-2}$ or $qx$ **providing one of these terms is obtained correctly.** |
| $\int 2x^{-2} + 4x\, dx = 2\times\frac{x^{-1}}{-1} + 4\times\frac{x^2}{2}\ (+c)$ | dM1A1 | dM1: $x^n \to x^{n+1}$ on any term. **Dependent on the first M.** A1: At least one term correct, simplified or un-simplified. Allow powers **and** coefficients to be un-simplified. |
| $= -\frac{2}{x} + 2x^2 + c$ | A1 | All correct and simplified including the $+ c$. Accept equivalents such as $-2x^{-1} + 2x^2 + c$ |
# Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\left(\frac{4}{\sqrt{x}}+k\right)dx = \int(4x^{-0.5}+k)dx = 4\frac{x^{0.5}}{0.5} + kx\ (+c)$ | M1A1 | M1: Integrates to obtain either $\alpha x^{0.5}$ or $kx$. A1: Correct integration (simplified or un-simplified). Allow powers **and** coefficients to be un-simplified e.g. $4\frac{x^{-0.5+1}}{0.5}$. There is no need for $+c$ |
| $\left[4\frac{x^{0.5}}{0.5}+kx\right]_{2}^{4} = 30 \Rightarrow (8\sqrt{4}+4k)-(8\sqrt{2}+2k) = 30$ | M1 | Substitutes both $x=4$ and $x=2$ into **changed** expression involving $k$, subtracts either way round and sets equal to 30. Condone poor use or omission of brackets when subtracting. |
| $2k + 16 - 8\sqrt{2} = 30 \Rightarrow k = 7 + 4\sqrt{2}$ | ddM1A1 | ddM1: Attempts to solve for $k$ from a linear equation in $k$. **Dependent upon both M's and need to have seen** $\int k\,dx \to kx$. A1: $7+4\sqrt{2}$ or exact equivalent e.g. $7+2^{2.5}$, $7+4\times 2^{0.5}$ |
7. (i) Find
$$\int \frac { 2 + 4 x ^ { 3 } } { x ^ { 2 } } \mathrm {~d} x$$
giving each term in its simplest form.\\
(ii) Given that $k$ is a constant and
$$\int _ { 2 } ^ { 4 } \left( \frac { 4 } { \sqrt { x } } + k \right) \mathrm { d } x = 30$$
find the exact value of $k$.
\hfill \mbox{\textit{Edexcel C12 2017 Q7 [9]}}