9 Express \(\frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\) in partial fractions and hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } } \frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \frac { 3 } { 2 } + \frac { 1 } { 3 }\).
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Question 9:
Answer Marks
Guidance
Answer Marks
Guidance
\(\dfrac{A}{1+2x} + \dfrac{B}{1-x} + \dfrac{C}{(1-x)^2}\) B1
or \(\dfrac{A}{1+2x} + \dfrac{Bx+C}{(1-x)^2}\); may be seen in later work
\(2 + x^2 \equiv A(1-x)^2 + B(1+2x)(1-x) + C(1+2x)\) M1
or \(A(1-x)^2 + (Bx+C)(1+2x)\)
\(A=1\), \(B=0\) and \(C=1\) A1A1A1
\(\int\left(\dfrac{1}{1+2x} + \dfrac{1}{(1-x)^2}\right)dx = a\ln(1+2x) + b(1-x)^{-1}\) M1*
\(a\) and \(b\) are non-zero constants
\(F(x) = \frac{1}{2}\ln(1+2x) + (1-x)^{-1}\) A1
their \(\dfrac{1}{2}\ln\dfrac{3}{2} + \dfrac{4}{3} - \left(\dfrac{1}{2}\ln 1 + 1\right)\) M1dep*
\(\dfrac{1}{2}\ln\dfrac{3}{2} + \dfrac{4}{3} - 0 - 1\) A1
and completion to given result
[9]
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# Question 9:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{A}{1+2x} + \dfrac{B}{1-x} + \dfrac{C}{(1-x)^2}$ | B1 | or $\dfrac{A}{1+2x} + \dfrac{Bx+C}{(1-x)^2}$; may be seen in later work | if B0M0, SC1 for $\dfrac{1}{1+2x}$ seen |
| $2 + x^2 \equiv A(1-x)^2 + B(1+2x)(1-x) + C(1+2x)$ | M1 | or $A(1-x)^2 + (Bx+C)(1+2x)$ | allow only sign errors, not algebraic errors |
| $A=1$, $B=0$ and $C=1$ | A1A1A1 | | |
| $\int\left(\dfrac{1}{1+2x} + \dfrac{1}{(1-x)^2}\right)dx = a\ln(1+2x) + b(1-x)^{-1}$ | M1* | $a$ and $b$ are non-zero constants | ignore extra terms |
| $F(x) = \frac{1}{2}\ln(1+2x) + (1-x)^{-1}$ | A1 | | |
| their $\dfrac{1}{2}\ln\dfrac{3}{2} + \dfrac{4}{3} - \left(\dfrac{1}{2}\ln 1 + 1\right)$ | M1dep* | | |
| $\dfrac{1}{2}\ln\dfrac{3}{2} + \dfrac{4}{3} - 0 - 1$ | A1 | and completion to given result | NB $\dfrac{1}{2}\ln\dfrac{3}{2} + \dfrac{1}{3}$ |
| **[9]** | | |
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9 Express $\frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }$ in partial fractions and hence show that $\int _ { 0 } ^ { \frac { 1 } { 4 } } \frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \frac { 3 } { 2 } + \frac { 1 } { 3 }$.
\hfill \mbox{\textit{OCR C4 2014 Q9 [9]}}