Moderate -0.3 This is a standard C4 partial fractions question with straightforward linear factors and a definite integral. Part (a) requires routine algebraic manipulation to find constants, and part (b) involves direct integration of logarithmic terms—both are textbook exercises with no novel problem-solving required. Slightly easier than average due to the simple denominator structure and clean numerical bounds.
3. (a) Express \(\frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) }\) in partial fractions.
(b) Hence find the exact value of \(\int _ { 2 } ^ { 6 } \frac { 5 x + 3 } { ( 2 x - 3 ) ( x + 2 ) } \mathrm { d } x\), giving your answer as a single logarithm.
Substituting \(x = -2\) or \(x = \frac{3}{2}\) and obtaining \(A\) or \(B\); or equating coefficients and solving a pair of simultaneous equations to obtain \(A\) or \(B\).
M1
\(A = 3, B = 1\)
A1, A1
If the cover-up rule is used, give M1 A1 for the first of \(A\) or \(B\) found, A1 for the second.
**(a)** $\frac{5x+3}{(2x-3)(x+2)} = \frac{A}{2x-3} + \frac{B}{x+2}$ | |
$5x + 3 = A(x+2) + B(2x-3)$ | |
Substituting $x = -2$ or $x = \frac{3}{2}$ and obtaining $A$ or $B$; or equating coefficients and solving a pair of simultaneous equations to obtain $A$ or $B$. | M1 |
$A = 3, B = 1$ | A1, A1 |
If the cover-up rule is used, give M1 A1 for the first of $A$ or $B$ found, A1 for the second. | |
**Total for (a): [3]**
**(b)** $\int\frac{5x+3}{(2x-3)(x+2)}dx = \frac{3}{2}\ln(2x-3) + \ln(x+2)$ | M1 A1ft |
$[\ldots]_2^6 = \frac{3}{2}\ln 9 + \ln 2 = \ln 54$ | M1 A1, cao A1 |
**Total for (b): [5]**
**Total: [8]**
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