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

9. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29\) and that \(C\) passes through the point \(P ( 2,6 )\),

1. find \(y\) in terms of \(x\).
2. Verify that \(C\) passes through the point \(( 4,0 )\).
3. Find an equation of the tangent to \(C\) at \(P\). The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
4. Calculate the exact \(x\)-coordinate of \(Q\).

5. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , x > 0\).

1. Show that, when \(x = 8\), the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(9 \sqrt { } 2\). The curve \(C\) passes through the point \(( 4,30 )\).
2. Using integration, find \(\mathrm { f } ( x )\).

4. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1 ,$$ and that \(y = 3\) when \(x = 0\), find the value of \(y\) when \(x = 2\).

1. Given that

$$y = 2 x ^ { \frac { 3 } { 2 } } - 1$$

1. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
2. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 y = k$$ where \(k\) is an integer to be found,
3. find $$\int y ^ { 2 } \mathrm {~d} x$$

5. Given that $$y = x + 5 + \frac { 3 } { \sqrt { x } }$$

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. find \(\int y \mathrm {~d} x\).

2. Find $$\int \left( 4 x ^ { 2 } - \sqrt { x } \right) \mathrm { d } x$$

8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-006_763_879_466_577} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
   3. Hence find the coordinates of the point \(P\) where the two tangents meet.
3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).

4

1. Write \(\sqrt { x }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
2. Hence express \(\sqrt { x } ( x - 1 )\) in the form \(x ^ { p } - x ^ { q }\).
3. Find \(\int \sqrt { x } ( x - 1 ) \mathrm { d } x\).
4. Hence show that \(\int _ { 1 } ^ { 2 } \sqrt { x } ( x - 1 ) \mathrm { d } x = \frac { 4 } { 15 } ( \sqrt { 2 } + 1 )\).

8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-5_778_901_461_571} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
   3. Hence find the coordinates of the point \(P\) where the two tangents meet.
3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).

5 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(P ( 4,0 )\).
The normal to the curve at \(P\) meets the \(y\)-axis at the point \(Q\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-3_526_629_916_813} The curve, defined for \(x \geqslant 0\), has equation $$y = 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } }$$

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (3 marks)
   2. Show that the gradient of the curve at \(P ( 4,0 )\) is - 2 .
   3. Find an equation of the normal to the curve at \(P ( 4,0 )\).
   4. Find the \(y\)-coordinate of \(Q\) and hence find the area of triangle \(O P Q\).
   5. The curve has a maximum point \(M\). Find the \(x\)-coordinate of \(M\).
2. 1. Find \(\int \left( 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
   2. Find the total area of the region bounded by the curve and the lines \(P Q\) and \(Q O\).

4 The diagram shows a sketch of the curves with equations \(y = 2 x ^ { \frac { 3 } { 2 } }\) and \(y = 8 x ^ { \frac { 1 } { 2 } }\). \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-3_433_720_1452_644} The curves intersect at the origin and at the point \(A\), where \(x = 4\).

1. 1. For the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\).
      (2 marks)
   2. Find an equation of the normal to the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\) at the point \(A\).
2. 1. Find \(\int 8 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
   2. Find the area of the shaded region bounded by the two curves.
3. Describe a single geometrical transformation that maps the graph of \(y = 2 x ^ { \frac { 3 } { 2 } }\) onto the graph of \(y = 2 ( x + 3 ) ^ { \frac { 3 } { 2 } }\).
   (2 marks)

2 At the point \(( x , y )\) on a curve, where \(x > 0\), the gradient is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 \sqrt { x ^ { 5 } } - 4$$

1. Write \(\sqrt { x ^ { 5 } }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
2. Find \(\int \left( 7 \sqrt { x ^ { 5 } } - 4 \right) \mathrm { d } x\).
3. Hence find the equation of the curve, given that the curve passes through the point \(( 1,3 )\).

5

1. Using the binomial expansion, or otherwise, express \(( 1 - x ) ^ { 3 }\) in ascending powers of \(x\).
2. Show that the expansion of $$( 1 + y ) ^ { 4 } - ( 1 - y ) ^ { 3 }$$ is $$7 y + p y ^ { 2 } + q y ^ { 3 } + y ^ { 4 }$$ where \(p\) and \(q\) are constants to be found.
3. Hence find \(\int \left[ ( 1 + \sqrt { x } ) ^ { 4 } - ( 1 - \sqrt { x } ) ^ { 3 } \right] \mathrm { d } x\), expressing each coefficient in its simplest form.

9 The diagram shows part of a curve crossing the \(x\)-axis at the origin \(O\) and at the point \(A ( 8,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-5_547_536_497_760} The curve has equation $$y = 12 x - 3 x ^ { \frac { 5 } { 3 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 8,0 )\) is \(y + 8 x = 64\).
3. Find \(\int \left( 12 x - 3 x ^ { \frac { 5 } { 3 } } \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve from \(O\) to \(A\) and the tangents \(O P\) and \(A P\).

2

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 1 } ^ { 5 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
2. 1. Find \(\int \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving the coefficient of each term in its simplest form.
   2. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\).

4 The diagram shows a curve \(C\) with equation \(y = \sqrt { x }\). The point \(O\) is the origin \(( 0,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-3_488_1136_1009_443} The region bounded by the curve \(C\), the \(x\)-axis and the vertical lines \(x = 1\) and \(x = 4\) is shown shaded in the diagram.

1. 1. Write \(\sqrt { x }\) in the form \(x ^ { p }\), where \(p\) is a constant.
   2. Find \(\int \sqrt { x } \mathrm {~d} x\).
   3. Hence find the area of the shaded region.
2. The point on \(C\) for which \(x = 4\) is \(P\). The tangent to \(C\) at the point \(P\) intersects the \(x\)-axis and the \(y\)-axis at the points \(A\) and \(B\) respectively.
   1. Find an equation for the tangent to the curve \(C\) at the point \(P\).
   2. Find the area of the triangle \(A O B\).
3. Describe the single geometrical transformation by which the curve with equation \(y = \sqrt { x - 1 }\) can be obtained from the curve \(C\).
4. Use the trapezium rule with four ordinates (three strips) to find an approximation for \(\int _ { 1 } ^ { 4 } \sqrt { x - 1 } \mathrm {~d} x\), giving your answer to three significant figures.

7 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7$$

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 4\).
      (1 mark)
   2. Write \(\frac { 16 } { x ^ { 2 } }\) in the form \(16 x ^ { k }\), where \(k\) is an integer.
   3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   4. Hence determine whether the point where \(x = 4\) is a maximum or a minimum, giving a reason for your answer.
2. The point \(P ( 1,8 )\) lies on the curve.
   1. Show that the gradient of the curve at the point \(P\) is 12 .
   2. Find an equation of the normal to the curve at \(P\).
3. 1. Find \(\int \left( 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7 \right) \mathrm { d } x\).
   2. Hence find the equation of the curve which passes through the point \(P ( 1,8 )\).

7

1. The expression \(\left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 }\) can be written in the form $$1 + \frac { p } { x ^ { 2 } } + \frac { q } { x ^ { 4 } } + \frac { 64 } { x ^ { 6 } }$$ By using the binomial expansion, or otherwise, find the values of the integers \(p\) and \(q\).
2. 1. Hence find \(\int \left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
   2. Hence find the value of \(\int _ { 1 } ^ { 2 } \left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).

4

1. The expression \(\left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 }\) can be written in the form $$1 + \frac { p } { x ^ { 2 } } + \frac { q } { x ^ { 4 } } - \frac { 1 } { x ^ { 6 } }$$ Find the values of the integers \(p\) and \(q\).
2. 1. Hence find \(\int \left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
   2. Hence find the value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).

      REFEREN	REFERENCE
      	\(\_\_\_\_\)
      	\includegraphics[max width=\textwidth, alt={}]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-5_40_1567_2637_272}

3

1. The expression \(\left( 2 + x ^ { 2 } \right) ^ { 3 }\) can be written in the form $$8 + p x ^ { 2 } + q x ^ { 4 } + x ^ { 6 }$$ Show that \(p = 12\) and find the value of the integer \(q\).
2. 1. Hence find \(\int \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
      (5 marks)
   2. Hence find the exact value of \(\int _ { 1 } ^ { 2 } \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
      (2 marks)

3

1. \(\quad\) Expand \(\left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 }\).
2. Hence find \(\int \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).
3. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).

6 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 4 } { x ^ { 2 } } - 11$$ The point \(P ( 2,1 )\) lies on the curve.

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 2\).
      (l mark)
   2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 2\).
   3. Hence state whether \(P\) is a maximum point or a minimum point, giving a reason for your answer.
2. Find the equation of the curve.

3

1. 1. Using the binomial expansion, or otherwise, express \(( 2 + y ) ^ { 3 }\) in the form \(a + b y + c y ^ { 2 } + y ^ { 3 }\), where \(a , b\) and \(c\) are integers.
   2. Hence show that \(\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }\) can be expressed in the form \(p + q x ^ { - 4 }\), where \(p\) and \(q\) are integers.
2. 1. Hence find \(\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).
   2. Hence find the value of \(\int _ { 1 } ^ { 2 } \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).

2

1. Find \(\int \left( 1 + 3 x ^ { \frac { 1 } { 2 } } + x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
2. 1. The expression \(( 1 + y ) ^ { 3 }\) can be written in the form \(1 + 3 y + n y ^ { 2 } + y ^ { 3 }\). Write down the value of the constant \(n\).
   2. Hence, or otherwise, expand \(( 1 + \sqrt { x } ) ^ { 3 }\).
3. Hence find the exact value of \(\int _ { 0 } ^ { 1 } ( 1 + \sqrt { x } ) ^ { 3 } \mathrm {~d} x\).

1

1. Find \(\int \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x\), where \(a\) is a constant.
2. Hence, given that \(\int _ { 1 } ^ { 3 } \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x = 16\), find the value of the constant \(a\).
   [0pt] [2 marks]