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

The gradient of a curve \(C\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( x ^ { 2 } + 3 \right) ^ { 2 } } { x ^ { 2 } } , x \neq 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } + 6 + 9 x ^ { - 2 }\). The point \(( 3,20 )\) lies on \(C\).
2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
   

3. Given that \(y = 2 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } , x \neq 0\), find

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

2. Find $$\int \left( 8 x ^ { 3 } + 6 x ^ { \frac { 1 } { 2 } } - 5 \right) d x$$ giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-03_40_38_2682_1914}

1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , \quad x > 0\), where

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x - \frac { 5 } { \sqrt { } x } - 2$$ Given that the point \(P ( 4,5 )\) lies on \(C\), find

1. \(\mathrm { f } ( x )\),
2. an equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6. Given that \(\frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\) can be written in the form \(6 x ^ { p } + 3 x ^ { q }\),

1. write down the value of \(p\) and the value of \(q\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\), and that \(y = 90\) when \(x = 4\),
2. find \(y\) in terms of \(x\), simplifying the coefficient of each term.

1. Find

$$\int \left( 6 x ^ { 2 } + \frac { 2 } { x ^ { 2 } } + 5 \right) \mathrm { d } x$$ giving each term in its simplest form.

7. The point \(P ( 4 , - 1 )\) lies on the curve \(C\) with equation \(y = \mathrm { f } ( x ) , x > 0\), and $$f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - \frac { 6 } { \sqrt { } x } + 3$$

1. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.
2. Find \(\mathrm { f } ( x )\).

10. A curve has equation \(y = \mathrm { f } ( x )\). The point \(P\) with coordinates \(( 9,0 )\) lies on the curve. Given that $$\mathrm { f } ^ { \prime } ( x ) = \frac { x + 9 } { \sqrt { } x } , \quad x > 0$$

1. find \(\mathrm { f } ( x )\).
2. Find the \(x\)-coordinates of the two points on \(y = \mathrm { f } ( x )\) where the gradient of the curve is equal to 10

2. Find $$\int \left( 10 x ^ { 4 } - 4 x - \frac { 3 } { \sqrt { } x } \right) \mathrm { d } x$$ giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-03_120_51_2599_1900}

9. $$f ^ { \prime } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) ^ { 2 } } { x ^ { 2 } } , \quad x \neq 0$$

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 9 x ^ { - 2 } + A + B x ^ { 2 }\),
   where \(A\) and \(B\) are constants to be found.
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\). Given that the point \(( - 3,10 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\),
3. find \(\mathrm { f } ( x )\).

4. Given that \(y = 2 x ^ { 5 } + \frac { 6 } { \sqrt { } x } , x > 0\), find in their simplest form

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

8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 37\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.

Given that \(y = 4 x ^ { 3 } - \frac { 5 } { x ^ { 2 } } , x \neq 0\), find in their simplest form

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

1. Find

$$\int \left( 2 x ^ { 4 } - \frac { 4 } { \sqrt { } x } + 3 \right) d x$$ giving each term in its simplest form.

1. Find

$$\int \left( 2 x ^ { 5 } - \frac { 1 } { 4 x ^ { 3 } } - 5 \right) \mathrm { d } x$$ giving each term in its simplest form.

7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$\mathrm { f } ^ { \prime } ( x ) = 30 + \frac { 6 - 5 x ^ { 2 } } { \sqrt { x } }$$ Given that the point \(P ( 4 , - 8 )\) lies on \(C\),

1. find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
2. Find \(\mathrm { f } ( x )\), giving each term in its simplest form.

1. Given

$$y = 3 \sqrt { x } - 6 x + 4 , \quad x > 0$$

1. find \(\int y \mathrm {~d} x\), simplifying each term.
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. Hence find the value of \(x\) such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where

$$f ^ { \prime } ( x ) = ( x - 3 ) ( 3 x + 5 )$$ Given that the point \(P ( 1,20 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying each term.
2. Show that $$f ( x ) = ( x - 3 ) ^ { 2 } ( x + A )$$ where \(A\) is a constant to be found.
3. Sketch the graph of \(C\). Show clearly the coordinates of the points where \(C\) cuts or meets the \(x\)-axis and where \(C\) cuts the \(y\)-axis.
   

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

10. For the curve \(C\) with equation \(y = \mathrm { f } ( x )\), \(\frac { d y } { d x } = x ^ { 3 } + 2 x - 7 .\)

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   (2)
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \geq 2\) for all values of \(x\).
   (1)
   Given that the point \(P ( 2,4 )\) lies on \(C\),
3. find \(y\) in terms of \(x\),
   (5)
4. find an equation for the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
   (5)
   1. continued

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph of the curves \(C _ { 1 }\) and \(C _ { 2 }\) The curves intersect when \(x = 2.5\) and when \(x = 4\) A table of values for some points on the curve \(C _ { 1 }\) is shown below, with \(y\) values given to 3 decimal places as appropriate. Using the trapezium rule with all the values of \(y\) in the table,

1. find, to 2 decimal places, an estimate for the area bounded by the curve \(C _ { 1 }\), the line with equation \(x = 2.5\), the \(x\)-axis and the line with equation \(x = 4\) The curve \(C _ { 2 }\) has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x + 9 \quad x > 0$$
2. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x + 9 \right) \mathrm { d } x\) The region \(R\), shown shaded in Figure 2, is bounded by the curves \(C _ { 1 }\) and \(C _ { 2 }\)
3. Use the answers to part (a) and part (b) to find, to one decimal place, an estimate for the area of the region \(R\).
   (3)

1. The curve \(C\) has equation

$$y = \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$ where \(k\) is a positive constant.

1. Show that $$\int _ { 1 } ^ { 16 } \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \mathrm {~d} x = a k ^ { 2 } + b k + \frac { 2046 } { 5 }$$ where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0e3b364c-151b-471d-acb6-01afb018fb75-26_645_670_904_699} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the curve \(C\) and the line \(l\).
   Given that \(l\) intersects \(C\) at the point \(A ( 1,9 )\) and at the point \(B ( 16 , q )\) where \(q\) is a constant,
2. show that \(k = 4\) The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and \(l\) Using the answers to parts (a) and (b),
3. find the area of region \(R\)

9. \begin{figure}[h] \end{figure} Figure 1 is a sketch of the curve \(C\) with equation $$y = 2 x ^ { \frac { 3 } { 2 } } ( 4 - x ) \quad x \geqslant 0$$ The point \(P\) is the stationary point of \(C\).

1. Find, using calculus, the \(x\) coordinate of \(P\). The region \(R _ { 1 }\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
   The region \(R _ { 2 }\), also shown shaded in Figure 1, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = k\), where \(k\) is a constant. Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
2. find, using calculus, the exact value of \(k\).

8. Solutions relying on calculator technology are not acceptable in this question.

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bfeb1724-9a00-4a36-9606-520395792b2b-22_556_822_351_561} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of a curve with equation $$y = \frac { 8 \sqrt { x } - 5 } { 2 x ^ { 2 } } \quad x > 0$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\) Find the exact area of \(R\).
2. Find the value of the constant \(k\) such that $$\int _ { - 3 } ^ { 6 } \left( \frac { 1 } { 2 } x ^ { 2 } + k \right) \mathrm { d } x = 55$$

1. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5$$ Find

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