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

1 It is given that \(\mathrm { f } ( x ) = 3 x - \frac { 5 } { x ^ { 3 } }\).
Find

1. \(\mathrm { f } ^ { \prime } ( x )\),
2. \(\mathrm { f } ^ { \prime \prime } ( x )\),
3. \(\int \mathrm { f } ( x ) \mathrm { d } x\).

1

1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 3 } - 3 x + \frac { 5 } { x ^ { 2 } } \right)\).
2. Find \(\int \left( 6 x ^ { 2 } - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x\).

7

1. Find \(\int x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x\).
2. Show that \(\int _ { 0 } ^ { 8 } x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x = a \times 2 ^ { 11 }\), where \(a\) is a positive integer to be determined.

4 Find \(\int \left( 9 x ^ { 2 } + \frac { 6 } { \sqrt { x } } \right) \mathrm { d } x\).

1 Find \(\int \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).

3 Show that the area of the region bounded by the curve \(y = 3 x ^ { - \frac { 3 } { 2 } }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis is \(6 - 2 \sqrt { 3 }\).

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

7 Find \(\int \left( 4 \sqrt { x } - \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x\). Answer all the questions
Section B (79 marks)

8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\). \includegraphics[max width=\textwidth, alt={}, center]{b83c4e3a-36a6-4ca9-b44f-489676ca86d4-06_469_802_411_603} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).

1. Find the area of the rectangle \(A B C D\).
2. 1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
   3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).

4

1. The function f is defined for all values of \(x\) by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\).
   1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(x + 1\).
   2. Given that \(\mathrm { f } ( 1 ) = 0\) and \(\mathrm { f } ( - 2 ) = 0\), write down two linear factors of \(\mathrm { f } ( x )\).
   3. Hence express \(x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-3_543_796_897_623}
   1. The curve intersects the \(y\)-axis at the point \(A\). Find the \(y\)-coordinate of \(A\).
   2. The curve crosses the \(x\)-axis when \(x = - 2\), when \(x = 1\) and also at the point \(B\). Use the results from part (a) to find the \(x\)-coordinate of \(B\).
3. 1. Find \(\int \left( x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8 \right) d x\).
   2. Hence find the area of the shaded region bounded by the curve and the \(x\)-axis.

8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\). \includegraphics[max width=\textwidth, alt={}, center]{81f6fc30-982b-47b5-bab3-076cc0cc6563-5_479_816_406_596} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).

1. Find the area of the rectangle \(A B C D\).
2. 1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
   3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).

6

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x - 10\).
   1. Use the Factor Theorem to show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) in the form \(( x - 2 ) \left( x ^ { 2 } + a x + b \right)\), where \(a\) and \(b\) are constants.
2. The curve \(C\) with equation \(y = x ^ { 3 } + x - 10\), sketched below, crosses the \(x\)-axis at the point \(Q ( 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{22c93dd5-d96a-4e31-8507-9c802e386231-3_444_547_1781_756}
   1. Find the gradient of the curve \(C\) at the point \(Q\).
   2. Hence find an equation of the tangent to the curve \(C\) at the point \(Q\).
   3. Find \(\int \left( x ^ { 3 } + x - 10 \right) \mathrm { d } x\).
   4. Hence find the area of the shaded region bounded by the curve \(C\) and the coordinate axes.

6 The curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{2f7a8e95-4994-4732-a9a4-306c7b6cad92-3_444_819_1434_609} The curve crosses the \(x\)-axis at the origin \(O\), and the point \(A ( 2 , - 6 )\) lies on the curve.

1. 1. Find the gradient of the curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) at the point \(A\).
   2. Hence find the equation of the normal to the curve at the point \(A\), giving your answer in the form \(x + p y + q = 0\), where \(p\) and \(q\) are integers.
2. 1. Find the value of \(\int _ { 0 } ^ { 2 } \left( 12 x ^ { 2 } - 19 x - 2 x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).

4 The curve sketched below passes through the point \(A ( - 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{889639d6-0a31-4569-8370-1e72291a0c47-3_538_734_365_662} The curve has equation \(y = 14 - x - x ^ { 4 }\) and the point \(P ( 1,12 )\) lies on the curve.

1. 1. Find the gradient of the curve at the point \(P\).
   2. Hence find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
2. 1. Find \(\int _ { - 2 } ^ { 1 } \left( 14 - x - x ^ { 4 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve \(y = 14 - x - x ^ { 4 }\) and the line \(A P\).
      (2 marks)

4 The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_447_752_438_653}

1. Given that \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\), find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find an equation of the tangent to the curve at the point \(A ( - 1,0 )\).
3. Verify that the point \(B\), where \(x = 1\), is a minimum point of the curve.
4. The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_451_757_1736_648}
   1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 5 } - 3 x ^ { 2 } + x + 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve between \(A\) and \(B\) and the line segments \(A O\) and \(O B\).

6 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of a curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 10 x ^ { 4 } - 6 x ^ { 2 } + 5$$ The curve passes through the point \(P ( 1,4 )\).

1. Find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
2. Find the equation of the curve.

4 The curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{3729de55-7139-4f41-8584-640f173c0e09-3_444_588_411_717} The curve touches the \(x\)-axis at the point \(A ( 1,0 )\) and cuts the \(x\)-axis at the point \(B\).

1. 1. Use the factor theorem to show that \(x - 3\) is a factor of $$\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$$
   2. Hence find the coordinates of \(B\).
2. The point \(M\), shown on the diagram, is a minimum point of the curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence determine the \(x\)-coordinate of \(M\).
3. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).
4. 1. Find \(\int \left( x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3 \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the coordinate axes.

5 The curve with equation \(y = x ^ { 3 } - 10 x ^ { 2 } + 28 x\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{f2c95d73-d3fe-48f7-af07-84f12bb06727-3_483_899_402_568} The curve crosses the \(x\)-axis at the origin \(O\) and the point \(A ( 3,21 )\) lies on the curve.

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence show that the curve has a stationary point when \(x = 2\) and find the \(x\)-coordinate of the other stationary point.
2. 1. Find \(\int \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x\).
   2. Hence show that \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x = 56 \frac { 1 } { 4 }\).
   3. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).

4 The curve with equation \(y = x ^ { 4 } - 8 x + 9\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-5_410_609_383_721} The point \(( 2,9 )\) lies on the curve.

1. 1. Find \(\int _ { 0 } ^ { 2 } \left( x ^ { 4 } - 8 x + 9 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(y = 9\).
2. The point \(A ( 1,2 )\) lies on the curve with equation \(y = x ^ { 4 } - 8 x + 9\).
   1. Find the gradient of the curve at the point \(A\).
   2. Hence find an equation of the tangent to the curve at the point \(A\).

6 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 3\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{c44c229e-44b2-4799-9c9c-bfccdd09d450-4_590_787_365_625} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and passes through the point \(B ( 1,2 )\).

1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 3 } - 2 x ^ { 2 } + 3 \right) \mathrm { d } x\).
   (5 marks)
2. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 2 x ^ { 2 } + 3\) and the line \(A B\).
   (3 marks)

5

1. 1. Express \(x ^ { 2 } - 3 x + 5\) in the form \(( x - p ) ^ { 2 } + q\).
   2. Hence write down the equation of the line of symmetry of the curve with equation \(y = x ^ { 2 } - 3 x + 5\).
2. The curve \(C\) with equation \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = x + 5\) intersect at the point \(A ( 0,5 )\) and at the point \(B\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-4_471_707_653_676}
   1. Find the coordinates of the point \(B\).
   2. Find \(\int \left( x ^ { 2 } - 3 x + 5 \right) \mathrm { d } x\).
   3. Find the area of the shaded region \(R\) bounded by the curve \(C\) and the line segment \(A B\).

6 A curve has equation \(y = x ^ { 5 } - 2 x ^ { 2 } + 9\). The point \(P\) with coordinates \(( - 1,6 )\) lies on the curve.

1. Find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
2. The point \(Q\) with coordinates \(( 2 , k )\) lies on the curve.
   1. Find the value of \(k\).
   2. Verify that \(Q\) also lies on the tangent to the curve at the point \(P\).
3. The curve and the tangent to the curve at \(P\) are sketched below. \includegraphics[max width=\textwidth, alt={}, center]{aa42b4fd-1e37-48b8-90ee-269916c4db2c-4_721_887_936_589}
   1. Find \(\int _ { - 1 } ^ { 2 } \left( x ^ { 5 } - 2 x ^ { 2 } + 9 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the tangent to the curve at \(P\).
      (3 marks)

6 The diagram shows a curve and a line which intersect at the points \(A , B\) and \(C\). \includegraphics[max width=\textwidth, alt={}, center]{f2124c89-79de-4758-b7b8-ff273345b9dd-7_574_844_349_609} The curve has equation \(y = x ^ { 3 } - x ^ { 2 } - 5 x + 7\) and the straight line has equation \(y = x + 7\). The point \(B\) has coordinates ( 0,7 ).

1. 1. Show that the \(x\)-coordinates of the points \(A\) and \(C\) satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
   2. Find the coordinates of the points \(A\) and \(C\).
2. Find \(\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x\).
3. Find the area of the shaded region \(R\) bounded by the curve and the line segment \(A B\).
   [0pt] [4 marks] \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0\). The point \(A ( 3 , - 2 )\) lies on the circle.

1. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0$$

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 \(\mathrm { f } ( x )\).
   

1. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$

1. Use integration to find \(y\) in terms of \(x\).
2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).