1.02y Partial fractions: decompose rational functions

11 A differential equation is given by \(2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ( 1 - y )\).

1. Express \(\frac { 2 } { y ( 1 - y ) }\) in partial fractions.
2. Hence show by integration that \(\frac { y ^ { 2 } } { ( 1 - y ) ^ { 2 } } = A \mathrm { e } ^ { x }\).
3. Given that \(x = 0\) when \(y = 2\), find the value of \(A\) and express \(y\) in terms of \(x\).

7

1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
2. Hence show that \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).

10

1. Express \(\frac { 1 } { ( k - 1 ) k ( k + 1 ) }\) in partial fractions.
2. Let \(S _ { n } = \sum _ { k = 3 } ^ { n } \frac { 1 } { ( k - 1 ) k ( k + 1 ) }\) for \(n \geqslant 3\). Use the method of differences to show that $$S _ { n } = \frac { 1 } { 12 } - \frac { 1 } { 2 n ( n + 1 ) }$$ and write down the limit of \(S _ { n }\) as \(n \rightarrow \infty\).
3. Given that \(k\) is a positive integer greater than 1 , explain why \(\frac { 1 } { k ^ { 3 } } < \frac { 1 } { ( k - 1 ) k ( k + 1 ) }\).
4. Show that \(\frac { 27 } { 24 } < \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ^ { 3 } } < \frac { 29 } { 24 }\).

8

1. Express \(\frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) }\) in the form \(\frac { A x + B } { x ^ { 2 } + 1 } + \frac { C } { x - 2 }\) where \(A , B\) and \(C\) are constants to be found.
2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 7 x ^ { 2 } - 12 x + 1 } { \left( x ^ { 2 } + 1 \right) ( x - 2 ) } \mathrm { d } x\).

10

1. Using partial fractions, find the general solution of the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y - y ^ { 3 } \text { for } 0 < y < 1$$ giving your solution in the form \(y = \mathrm { f } ( x )\).
2. Determine \(\lim _ { x \rightarrow - \infty } \mathrm { f } ( x )\) and \(\lim _ { x \rightarrow + \infty } \mathrm { f } ( x )\).

7

1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x - 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
2. Hence show that \(\equiv \frac { 5 x - 1 } { \overline { 2 } } \frac { 8 x - 1 ) ( x + 1 ) } { ( 2 x - \ln 24 \text {. } }\)

7

1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
2. Hence show that \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).

7 Express \(\frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in the form \(\frac { A } { x + 2 } + \frac { B x + C } { x ^ { 2 } + 1 }\) where the numerical values of \(A , B\) and \(C\) are to be found. Hence show that \(\int _ { 0 } ^ { 1 } \frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) } \mathrm { d } x = \ln 3 - \frac { 5 } { 2 } \ln 2\).

a) Express \(\frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } }\) in terms of partial fractions. b) Find \(\int \frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } } \mathrm {~d} x\).

The diagram below shows a plan of the patio Eric wants to build. The walls \(O A\) and \(O C\) are perpendicular. The straight line \(A B\) is of length 4 m and is perpendicular to \(O A\). The shape \(O B C\) is a sector of a circle with centre \(O\) and radius OC.
The angle \(B O C\) is \(\frac { \pi } { 3 }\) radians. Calculate the area of the patio \(O A B C\). Give your answer correct to 2 decimal places. The sum to infinity of a geometric series with first term \(a\) and common ratio \(r\) is 120 . The sum to infinity of another geometric series with first term \(a\) and common ratio \(4 r ^ { 2 }\) is \(112 \frac { 1 } { 2 }\). Find the possible values of \(r\) and the corresponding values of \(a\). The function \(f ( x )\) is defined by $$f ( x ) = \frac { 6 x + 4 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) }$$ a) Express \(f ( x )\) in terms of partial fractions.
b) Find \(\int \frac { 3 x + 2 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) } \mathrm { d } x\), giving your answer in the form \(a \ln | g ( x ) |\), where \(a\) is a real number and \(g ( x )\) is a function of \(x\). Geraint opens a savings account. He deposits \(\pounds 10\) in the first month. In each subsequent month, the amount he deposits is 20 pence greater than the amount he deposited in the previous month.
a) Find the amount that Geraint deposits into the savings account in the 12th month.
b) Determine the number of months it takes for the total amount in the savings account to reach \(\pounds 954\).
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The diagram below shows a sketch of the curves \(y = x ^ { 2 }\) and \(y = 8 \sqrt { x }\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-3_508_869_2094_623} Find the area of the region bounded by the two curves.

Express \(\frac{6x^2 - 2x + 2}{(x - 1)(2x + 1)}\) in partial fractions. [5]

Let \(\text{f}(x) = \frac{5a}{(2x - a)(3a - x)}\), where \(a\) is a positive constant.

1. Express f\((x)\) in partial fractions. [3]
2. Hence show that \(\int_a^{2a} \text{f}(x) \, dx = \ln 6\). [4]

A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40\pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8\pi r\). The balloon remains a sphere at all times.

1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac{dr}{dt} = \frac{50 - r}{5r^2}.$$ [3]
2. Find the quotient and remainder when \(5r^2\) is divided by \(50 - r\). [3]
3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\). [6]
4. Find the value of \(t\) when the radius of the balloon is 12. [1]

Let \(f(x) = \frac{2e^{2x}}{e^{2x} - 3e^x + 2}\).

1. Find \(f'(x)\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = f(x)\). [5]
2. Use the substitution \(u = e^x\) and partial fractions to find the exact value of \(\int_{\ln 5} f(x) dx\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form. [9]

Let \(\mathrm{f}(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}\).

1. Express \(\mathrm{f}(x)\) in partial fractions. [5]
2. Hence obtain the expansion of \(\mathrm{f}(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

Let \(f(x) = \frac{2x^2 - 7x - 1}{(x-2)(x^2+3)}\).

1. Express \(f(x)\) in partial fractions. [5]
2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

Let \(\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}\).

1. Express \(\text{f}(x)\) in the form \(A + \frac{B}{x + 2} + \frac{C}{2x - 1}\). [4]
2. Hence show that \(\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)\). [5]

Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).

1. Express \(f(x)\) in partial fractions. [5]
2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

Let \(f(x) = \frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}\).

1. Express \(f(x)\) in partial fractions. [5]
2. Hence, showing all necessary working, show that \(\int_{-1}^0 f(x) dx = 1 + \frac{1}{2}\ln\left(\frac{4}{3}\right)\). [5]

The line \(y = 2x + 1\) is an asymptote of the curve \(C\) with equation $$y = \frac{x^2 + 1}{ax + b}.$$

1. Find the values of the constants \(a\) and \(b\). [3]
2. State the equation of the other asymptote of \(C\). [1]
3. Sketch \(C\). [Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.] [3]

Express \(\frac{x}{(x+1)(x+3)} + \frac{x+12}{x^2-9}\) as a single fraction in its simplest form. [6]