Question
Let f(x) \(\frac{6x^{2}+8x+9}{(2-x)(3+2x)^{2}}\)
(i) Express f(x) in partial fractions.
(ii) Hence, showing all necessary working, show that \(\int_{-1}^{0}f(x)dx=1+\frac{1}{2}In (\frac{3}{4}).\)
Answer/Explanation
(i) State or imply the form
Use a correct method to find a constant
Obtain one of A = 1, B = – 1, C = 3
Obtain a second value
Obtain the third value
[Mark the form \(\frac{A}{2-x}+\frac{Dx+E}{(3+2x)^{2}} \), where A = 1, D = – 2 and E = 0, B1M1A1A1A1 as above.]
(ii) Integrate and obtain terms\( -In(2-x)-\frac{1}{2} In (3+2x)-\frac{3}{2(3+2x)^{2}}\)
Substitute correctly in an integral with terms a ln (2 – x),
b ln (3 + 2x) and c / (3 + 2x) where abc ≠ 0
Obtain the given answer after full and correct working
[Correct integration of the
A, D, E form gives an extra constant term if integration by
parts is used for the second partial fraction.]
Question
Let \(fx =\frac{x^{2}+x+6}{x^{2}(x+2)}\)
(i) Express fx in partial fractions.
(ii) Hence, showing full working, show that the exact value of \(\int _{1}^{4f(x)dx is \frac{9}{4}}\)<
Answer/Explanation
(i) State or imply the form \(\frac{A}{x}+\frac{B}){x^{2}}+\frac{c}{x+2}\ Use a correct method for finding a constant
Obtain one of A = – 1, B = 3, C = 2
Obtain a second value
Obtain the third value
(ii) Integrate and obtain terms In \( x-\frac{3}{x}+2 In(x+2)\) Substitute limits correctly in an integral with terms a In x\( \frac{b}{x}\) and c x ln 2 ( + ) , where abc
≠ 0
Obtain\( \frac{9}{4}\) following full and exact working
Question
Let f(x )=\(\frac{16-17x}{(2+x)(3-x)^{2}}\)
(i) Express f(x )in partial fractions.
(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in \(x^{2}\)
Answer/Explanation
(i)State or imply the form \(\frac{A}{2+x}+\frac{B}{3-x}+\frac{C}{\left ( 3-x \right )^{2}}\)
Use a correct method to obtain a constant
Obtain one of A=2,B=2,C=-7
Obtain a second value
Obtain the third value
(ii) Use a correct method to find the first two terms of the expansion of \(\left ( 2+x \right )^{-1},\left ( 3-x \right )^{-1}\) or \(\left ( 3-x \right )^{-2}\), or equibvalent ,e.g.\(\left ( 1+\frac{1}{2}x \right )^{-1}\)
Obtain correct Unsimplified expansions up to the term in\( x^{2}\) of each partial fraction.
Obtain final answer \(\frac{8}{9}-\frac{43}{54}x+\frac{7}{108}x^{2}\)