1.08b Integrate x^n: where n != -1 and sums

7. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-3_499_721_248_552} The diagram shows part of the curve \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { 1 - 8 x ^ { 3 } } { x ^ { 2 } } , x \neq 0\).

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
3. Find the area of the shaded region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = 2\).

8 The gradient of a curve is \(6 x ^ { 2 } + 12 x ^ { \frac { 1 } { 2 } }\). The curve passes through the point \(( 4,10 )\). Find the equation of the curve.

11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - x ^ { 2 }\). The curve passes through the point \(( 6,1 )\). Find the equation of the curve.

3 A farmer digs ditches for flood relief. He experiments with different cross-sections. Assume that the surface of the ground is horizontal.

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-3_432_640_410_745} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
   \end{figure} Fig. 9.1 shows the cross-section of one ditch, with measurements in metres. The width of the ditch is 1.2 m and Fig. 9.1 shows the depth every 0.2 m across the ditch. Use the trapezium rule with six intervals to estimate the area of cross-section. Hence estimate the volume of water that can be contained in a 50 -metre length of this ditch.
2. Another ditch is 0.9 m wide, with cross-section as shown in Fig. 9.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-3_579_813_1336_656} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} With \(x\) - and \(y\)-axes as shown in Fig. 9.2, the curve of the ditch may be modelled closely by \(y = 3.8 x ^ { 4 } - 6.8 x ^ { 3 } + 7.7 x ^ { 2 } - 4.2 x\).
   (A) The actual ditch is 0.6 m deep when \(x = 0.2\). Calculate the difference between the depth given by the model and the true depth for this value of \(x\).
   (B) Find \(\int \left( 3.8 x ^ { 4 } - 6.8 x ^ { 3 } + 7.7 x ^ { 2 } - 4.2 x \right) \mathrm { d } x\). Hence estimate the volume of water that can be contained in a 50 -metre length of this ditch.

6 \includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-3_483_956_264_593} The diagram shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).

1. Find the exact area of the shaded region.
2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed, simplifying your answer.

5

1. Find \(\int ( 3 x + 7 ) ^ { 9 } \mathrm {~d} x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_537_881_402_671} The diagram shows the curve \(y = \frac { 1 } { 2 \sqrt { x } }\). The shaded region is bounded by the curve and the lines \(x = 3 , x = 6\) and \(y = 0\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced, simplifying your answer.

8 Fig. 8 shows part of the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \text { for } x \geqslant 0 .$$ \begin{figure}[h] \end{figure}

1. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the origin and at the point \(( \ln 2,1 )\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \ln ( 1 + \sqrt { x } )\) for \(x \geqslant 0\).
2. Show that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are inverse functions. Hence sketch the graph of \(y = \mathrm { g } ( x )\). Write down the gradient of the curve \(y = \mathrm { g } ( x )\) at the point \(( 1 , \ln 2 )\).
3. Show that \(\int \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { x } + x + c\). Hence evaluate \(\int _ { 0 } ^ { \ln 2 } \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \mathrm {~d} x\), giving your answer in an exact form.
4. Using your answer to part (iii), calculate the area of the region enclosed by the curve \(y = \mathrm { g } ( x )\), the \(x\)-axis and the line \(x = 1\).

6

1. Find \(\int ( 2 x - 3 ) ^ { 7 } \mathrm {~d} x\).
2. Use the substitution \(u = x ^ { 2 } + 1\), or otherwise, to find \(\int _ { 1 } ^ { 2 } x \left( x ^ { 2 } + 1 \right) ^ { 3 } \mathrm {~d} x\).

2 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), which has a \(y\)-intercept at \(\mathrm { P } ( 0,3 )\), a minimum point at \(\mathrm { Q } ( 1,2 )\), and an asymptote \(x = - 1\). \begin{figure}[h] \end{figure}

1. Find the coordinates of the images of the points P and Q when the curve \(y = \mathrm { f } ( x )\) is transformed to
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( x + 1 ) + 2\). You are now given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { x + 1 } , x \neq - 1\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence find the coordinates of the other turning point on the curve \(y = \mathrm { f } ( x )\).
3. Show that \(\mathrm { f } ( x - 1 ) = x - 2 + \frac { 4 } { x }\).
4. Find \(\int _ { a } ^ { b } \left( x - 2 + \frac { 4 } { x } \right) \mathrm { d } x\) in terms of \(a\) and \(b\). Hence, by choosing suitable values for \(a\) and \(b\), find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).

4

1. Solve the equation $$\sinh x + 4 \cosh x = 8$$ giving the answers in an exact logarithmic form.
2. Find the exact value of \(\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x } \sinh x \mathrm {~d} x\).
3. 1. Differentiate \(\operatorname { arsinh } \left( \frac { 2 } { 3 } x \right)\) with respect to \(x\).
   2. Use integration by parts to show that \(\int _ { 0 } ^ { 2 } \operatorname { arsinh } \left( \frac { 2 } { 3 } x \right) \mathrm { d } x = 2 \ln 3 - 1\).

1. Find

$$\int \left( 8 x ^ { 3 } + 4 \right) d x$$ giving each term in its simplest form.

10. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point (4,25). Given that $$f ^ { \prime } ( x ) = \frac { 3 } { 8 } x ^ { 2 } - 10 x ^ { - \frac { 1 } { 2 } } + 1 , \quad x > 0$$

1. find \(\mathrm { f } ( x )\), simplifying each term.
2. Find an equation of the normal to the curve at the point ( 4,25 ). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.

1 Find

1. \(\int \left( x ^ { 3 } + 8 x - 5 \right) \mathrm { d } x\),
2. \(\int 12 \sqrt { x } \mathrm {~d} x\).

6

1. Find \(\int \frac { x ^ { 3 } + 3 x ^ { \frac { 1 } { 2 } } } { x } \mathrm {~d} x\).
2. 1. Find, in terms of \(a\), the value of \(\int _ { 2 } ^ { a } 6 x ^ { - 4 } \mathrm {~d} x\), where \(a\) is a constant greater than 2 .
   2. Deduce the value of \(\int _ { 2 } ^ { \infty } 6 x ^ { - 4 } \mathrm {~d} x\).

7

1. Find \(\int \left( x ^ { 2 } + 4 \right) ( x - 6 ) \mathrm { d } x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-4_449_551_349_758} The diagram shows the curve \(y = 6 x ^ { \frac { 3 } { 2 } }\) and part of the curve \(y = \frac { 8 } { x ^ { 2 } } - 2\), which intersect at the point \(( 1,6 )\). Use integration to find the area of the shaded region enclosed by the two curves and the \(x\)-axis.

3 A curve has an equation which satisfies \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k x ( 2 x - 1 )\) for all values of \(x\). The point \(P ( 2,7 )\) lies on the curve and the gradient of the curve at \(P\) is 9 .

1. Find the value of the constant \(k\).
2. Find the equation of the curve.

9 The positive constant \(a\) is such that \(\int _ { a } ^ { 2 a } \frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 4 } { x ^ { 2 } } \mathrm {~d} x = 0\).

1. Show that \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\).
2. Show that \(a = 1\) is a root of \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\), and hence find the other possible value of \(a\), giving your answer in simplified surd form.

4

1. Find the binomial expansion of \(\left( x ^ { 2 } - 5 \right) ^ { 3 }\), simplifying the terms.
2. Hence find \(\int \left( x ^ { 2 } - 5 \right) ^ { 3 } \mathrm {~d} x\).

6

1. Use integration to find the exact area of the region enclosed by the curve \(y = x ^ { 2 } + 4 x\), the \(x\)-axis and the lines \(x = 3\) and \(x = 5\).
2. Find \(\int ( 2 - 6 \sqrt { y } ) \mathrm { d } y\).
3. Evaluate \(\int _ { 1 } ^ { \infty } \frac { 8 } { x ^ { 3 } } \mathrm {~d} x\).

2

1. Find \(\int \left( 6 x ^ { \frac { 1 } { 2 } } - 1 \right) \mathrm { d } x\).
2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { \frac { 1 } { 2 } } - 1\) and which passes through the point \(( 4,17 )\).

2

1. Find \(\int \left( x ^ { 2 } - 2 x + 5 \right) \mathrm { d } x\).
2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 2 x + 5\) and which passes through the point \(( 3,11 )\).

6

1. Use the trapezium rule, with 2 strips each of width 4 , to show that an approximate value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\) is \(32 + 16 \sqrt { 5 }\).
2. Use a sketch graph to explain why the actual value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\) is greater than \(32 + 16 \sqrt { 5 }\).
3. Use integration to find the exact value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\).

4

1. Find \(\int \left( 5 x ^ { 3 } - 6 x + 1 \right) \mathrm { d } x\).
2. 1. Find \(\int 24 x ^ { - 3 } \mathrm {~d} x\).
   2. Given that \(\int _ { a } ^ { \infty } 24 x ^ { - 3 } \mathrm {~d} x = 3\), find the value of the positive constant \(a\).

7 \includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-3_519_611_1087_712} The diagram shows the curve \(y = x ^ { \frac { 3 } { 2 } } - 1\), which crosses the \(x\)-axis at \(( 1,0 )\), and the tangent to the curve at the point \(( 4,7 )\).

1. Show that \(\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) \mathrm { d } x = 9 \frac { 2 } { 5 }\).
2. Hence find the exact area of the shaded region enclosed by the curve, the tangent and the \(x\)-axis.

6

1. Find the binomial expansion of \(\left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 }\), simplifying the terms.
2. Hence find \(\int \left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 } \mathrm {~d} x\).