1.08j Integration using partial fractions

4 Let \(f ( x ) = \frac { 15 - 6 x } { ( 1 + 2 x ) ( 4 - x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(\ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 9 x } { ( 3 x - 1 ) \left( x ^ { 2 } + 3 \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int _ { 1 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in a simplified exact form.

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

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find the exact value of \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.

9 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 17 x - 17 } { ( 1 + 2 x ) ( 2 - x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 5 } { 2 } - \ln 72\). \includegraphics[max width=\textwidth, alt={}, center]{60bb482b-fa41-42ea-a112-62851e5a19aa-16_524_725_269_696} The diagram shows the curve \(y = ( x + 5 ) \sqrt { 3 - 2 x }\) and its maximum point \(M\).

11 Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + x + 11 } { \left( 4 + x ^ { 2 } \right) ( 1 + x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
2. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln 54 - \frac { 1 } { 8 } \pi\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

10 Let \(f ( x ) = \frac { 36 a ^ { 2 } } { ( 2 a + x ) ( 2 a - x ) ( 5 a - 2 x ) }\), where \(a\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find the exact value of \(\int _ { - a } ^ { a } f ( x ) d x\), giving your answer in the form plnq+rlns where \(p\) and \(r\) are integers and \(q\) and \(s\) are prime numbers.

9 Let \(\mathrm { f } ( x ) = \frac { 7 x + 18 } { ( 3 x + 2 ) \left( x ^ { 2 } + 4 \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find the exact value of \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).

10 Let \(\mathrm { f } ( x ) = \frac { 4 - x + x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Find the exact value of \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer as a single logarithm.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 Let \(\mathrm { f } ( x ) = \frac { 5 - x + 6 x ^ { 2 } } { ( 3 - x ) \left( 1 + 3 x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\), simplifying your answer.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Find the exact value of \(\int _ { 0 } ^ { 6 } \frac { x ( x + 1 ) } { x ^ { 2 } + 4 } \mathrm {~d} x\).

8. (i) Find, using algebraic integration, the exact value of $$\int _ { 3 } ^ { 42 } \frac { 2 } { 3 x - 1 } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) $$\mathrm { h } ( x ) = \frac { 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 1 } { ( x - 1 ) ^ { 2 } } \quad x > 1$$ Given \(\mathrm { h } ( x ) = A x + B + \frac { C } { ( x - 1 ) ^ { 2 } }\) where \(A , B\) and \(C\) are constants to be found, find $$\int \mathrm { h } ( x ) \mathrm { d } x$$ \includegraphics[max width=\textwidth, alt={}, center]{1c700103-ecab-4a08-b411-3f445ed88885-26_2258_47_312_1985}

4. $$f ( x ) = \frac { 2 x ^ { 4 } + 15 x ^ { 3 } + 35 x ^ { 2 } + 21 x - 4 } { ( x + 3 ) ^ { 2 } } \quad x \in \mathbb { R } \quad x > - 3$$

1. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that $$\mathrm { f } ( x ) = A x ^ { 2 } + B x + C + \frac { D } { ( x + 3 ) ^ { 2 } }$$
2. Hence find, $$\int \mathrm { f } ( x ) \mathrm { d } x$$

9. (a) Given that $$\frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \equiv x ^ { 2 } + P + \frac { Q } { x - 4 } \quad x > - 3$$ find the value of the constant \(P\) and show that \(Q = 5\) The curve \(C\) has equation \(y = \mathrm { g } ( x )\), where $$g ( x ) = \frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \quad - 3 < x < 3.5 \quad x \in \mathbb { R }$$ (b) Find the equation of the tangent to \(C\) at the point where \(x = 2\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\).
The region \(R\), shown shaded in Figure 4, is bounded by \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 2\) (c) Find the exact area of \(R\), writing your answer in the form \(a + b \ln 2\), where \(a\) and \(b\) are constants to be found. \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-31_2255_50_314_34}
\includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-32_106_113_2524_1832}

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$f ( x ) = \frac { 2 x ^ { 3 } - 4 x - 15 } { x ^ { 2 } + 3 x + 4 }$$

1. Show that $$f ( x ) \equiv A x + B + \frac { C ( 2 x + 3 ) } { x ^ { 2 } + 3 x + 4 }$$ where \(A , B\) and \(C\) are integers to be found.
2. Hence, find $$\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

3. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$

1. find the values of the constants \(A , B , C\) and \(D\).
2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

9. \begin{figure}[h] \end{figure} The curve \(C\) has parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 4 } { t ^ { 2 } } \quad t > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = \ln 3\) and \(x = \ln 5\)

1. Show that the area of \(R\) is given by the integral $$\int _ { 1 } ^ { 3 } \frac { 4 } { t ^ { 2 } ( t + 2 ) } \mathrm { d } t$$
2. Hence find an exact value for the area of \(R\). Write your answer in the form ( \(a + \ln b\) ), where \(a\) and \(b\) are rational numbers.
3. Find a cartesian equation of the curve \(C\) in the form \(y = \mathrm { f } ( x )\).

1. (a) Express \(\frac { 9 ( 4 + x ) } { 16 - 9 x ^ { 2 } }\) in partial fractions.

Given that $$\mathrm { f } ( x ) = \frac { 9 ( 4 + x ) } { 16 - 9 x ^ { 2 } } , \quad x \in \mathbb { R } , \quad - \frac { 4 } { 3 } < x < \frac { 4 } { 3 }$$ (b) express \(\int \mathrm { f } ( x ) \mathrm { d } x\) in the form \(\ln ( \mathrm { g } ( x ) )\), where \(\mathrm { g } ( x )\) is a rational function.

11. (a) Given $$\frac { 9 } { t ^ { 2 } ( t - 3 ) } \equiv \frac { A } { t } + \frac { B } { t ^ { 2 } } + \frac { C } { ( t - 3 ) }$$ find the value of the constants \(A , B\) and \(C\).
(b) $$I = \int _ { 4 } ^ { 12 } \frac { 9 } { t ^ { 2 } ( t - 3 ) } \mathrm { d } t$$ Find the exact value of \(I\), giving your answer in the form \(\ln ( a ) - b\), where \(a\) and \(b\) are positive constants. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = 2 \ln ( t - 3 ) , \quad y = \frac { 6 } { t } \quad t > 3$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 2 \ln 9\) The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
(c) Show that the exact volume of the solid generated is $$k \times I$$ where \(k\) is a constant to be found.

8. Use partial fractions, and integration, to find the exact value of \(\int _ { 3 } ^ { 4 } \frac { 2 x ^ { 2 } - 3 } { x ( x - 1 ) } \mathrm { d } x\) Write your answer in the form \(a + \ln b\), where \(a\) is an integer and \(b\) is a rational constant.

5. $$f ( x ) = \frac { 4 x ^ { 2 } + 5 x + 3 } { ( x + 2 ) ( 1 - x ) ^ { 2 } } \equiv \frac { A } { ( x + 2 ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$$

1. Find the values of the constants \(A\), \(B\) and \(C\).
2. 1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
   2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\), writing your answer in the form \(p + \ln q\), where \(p\) and \(q\) are constants.
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4. $$\mathrm { f } ( x ) = \frac { 4 - 4 x } { x ( x - 2 ) ^ { 2 } } \quad x > 2$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
3. Find $$\int _ { 3 } ^ { 5 } f ( x ) d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are rational numbers to be found.

1. Given that

$$\frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \equiv \frac { A } { x - 2 } + \frac { B } { 2 x + 1 } + \frac { C } { ( 2 x + 1 ) ^ { 2 } }$$

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find the exact value of $$\int _ { 7 } ^ { 12 } \frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in the form \(p \ln q + r\) where \(p\), \(q\) and \(r\) are rational numbers.

3. (a) Express \(\frac { 5 } { ( x - 1 ) ( 3 x + 2 ) }\) in partial fractions.
(b) Hence find \(\int \frac { 5 } { ( x - 1 ) ( 3 x + 2 ) } \mathrm { d } x\), where \(x > 1\).
(c) Find the particular solution of the differential equation $$( x - 1 ) ( 3 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = 5 y , \quad x > 1$$ for which \(y = 8\) at \(x = 2\). Give your answer in the form \(y = \mathrm { f } ( x )\).

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.

4. $$\frac { 2 \left( 4 x ^ { 2 } + 1 \right) } { ( 2 x + 1 ) ( 2 x - 1 ) } \equiv A + \frac { B } { ( 2 x + 1 ) } + \frac { C } { ( 2 x - 1 ) } .$$

1. Find the values of the constants \(A , B\) and \(C\).
2. Hence show that the exact value of \(\int _ { 1 } ^ { 2 } \frac { 2 \left( 4 x ^ { 2 } + 1 \right) } { ( 2 x + 1 ) ( 2 x - 1 ) } \mathrm { d } x\) is \(2 + \ln k\), giving the value of the constant \(k\).