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

7. \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = x ^ { 2 } - 5 x + 4\). It cuts the \(x\)-axis at the points \(L\) and \(M\) as shown in Figure 2.

1. Find the coordinates of the point \(L\) and the point \(M\).
2. Show that the point \(N ( 5,4 )\) lies on \(C\).
3. Find \(\int \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\). The finite region \(R\) is bounded by \(L N , L M\) and the curve \(C\) as shown in Figure 2.
4. Use your answer to part (c) to find the exact value of the area of \(R\).
   \section*{LU}

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = ( x + 1 ) ( x - 5 )$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\).

1. Write down the \(x\)-coordinates of \(A\) and \(B\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
2. Use integration to find the area of \(R\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the curve with equation $$y = \frac { 16 } { x ^ { 2 } } - \frac { x } { 2 } + 1 , \quad x > 0$$ The finite region \(R\), bounded by the lines \(x = 1\), the \(x\)-axis and the curve, is shown shaded in Figure 1. The curve crosses the \(x\)-axis at the point \(( 4,0 )\).

1. Complete the table with the values of \(y\) corresponding to \(x = 2\) and 2.5

   \(x\)	1	1.5	2	2.5	3	3.5	4
   \(y\)	16.5	7.361			1.278	0.556	0

2. Use the trapezium rule with all the values in the completed table to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
3. Use integration to find the exact value for the area of \(R\).

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).

1. Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
   The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
3. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
4. Hence calculate the exact area of \(R\).

Evaluate \(\int _ { 1 } ^ { 8 } \frac { 1 } { \sqrt { } x } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = 10 + 8 x + x ^ { 2 } - x ^ { 3 }\).
The curve has a maximum turning point \(A\).

1. Using calculus, show that the \(x\)-coordinate of \(A\) is 2 . The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(y\)-axis and the line from \(O\) to \(A\), where \(O\) is the origin.
2. Using calculus, find the exact area of \(R\).

1. Use calculus to find the value of

$$\int _ { 1 } ^ { 4 } ( 2 x + 3 \sqrt { } x ) d x$$

1. Use integration to find

$$\int _ { 1 } ^ { \sqrt { 3 } } \left( \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 3 x ^ { 2 } } \right) \mathrm { d } x$$ giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are constants to be determined.

6.

1. Find $$\int 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) \mathrm { d } x$$ giving each term in its simplest form. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8a7593c3-4f0b-4351-afae-7bd98cfc351d-10_401_1002_543_470} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) , \quad x \geqslant 0$$ The curve \(C\) starts at the origin and crosses the \(x\)-axis at the point \(( 4,0 )\). The area, shown shaded in Figure 2, consists of two finite regions and is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 9\)
2. Use your answer from part (a) to find the total area of the shaded regions.

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 3 x - x ^ { \frac { 3 } { 2 } } , \quad x \geqslant 0$$ The finite region \(S\), bounded by the \(x\)-axis and the curve, is shown shaded in Figure 3.

1. Find $$\int \left( 3 x - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x$$
2. Hence find the area of \(S\).

1. Find

$$\int \frac { x ^ { 2 } - 5 } { 2 x ^ { 3 } } \mathrm {~d} x \quad x > 0$$ giving your answer in simplest form.

5.

1. Find, by algebraic integration, the exact value of $$\int _ { 2 } ^ { 4 } \frac { 8 } { ( 2 x - 3 ) ^ { 3 } } d x$$
2. Find, in simplest form, $$\int x \left( x ^ { 2 } + 3 \right) ^ { 7 } d x$$

9. \begin{figure}[h] \end{figure} Diagram not drawn to scale

1. Find $$\int \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } d x$$ Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 12 } { ( 2 x - 1 ) } \quad 1 \leqslant x \leqslant 5$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the line with equation \(x = 1\), the curve with equation \(y = \mathrm { f } ( x )\) and the line with equation \(y = \frac { 4 } { 3 }\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Find the exact value of the volume of the solid generated, giving your answer in its simplest form.
   \section*{Leave
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2.

1. Find \(\int \frac { 4 x + 3 } { x } \mathrm {~d} x , \quad x > 0\)
2. Given that \(y = 25\) at \(x = 1\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 4 x + 3 ) y ^ { \frac { 1 } { 2 } } } { x } \quad x > 0 , y > 0$$ giving your answer in the form \(y = [ \mathrm { g } ( x ) ] ^ { 2 }\).
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5.

1. Find $$\int \left( ( 3 x + 5 ) ^ { 9 } + \mathrm { e } ^ { 5 x } \right) \mathrm { d } x$$
2. Given that \(b\) is a constant greater than 2 , and $$\int _ { 2 } ^ { b } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln ( \sqrt { 6 } )$$ use integration to find the value of \(b\).

7.

1. Find $$\int ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } d x$$ giving your answer in its simplest form. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-13_727_1177_596_370} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = ( 2 x - 1 ) ^ { \frac { 3 } { 4 } } , \quad x \geqslant \frac { 1 } { 2 }$$ The curve \(C\) cuts the line \(y = 8\) at the point \(P\) with coordinates \(( k , 8 )\), where \(k\) is a constant.
2. Find the value of \(k\). The finite region \(S\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(y = 8\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Find the exact value of the volume of the solid generated.

3. (i) Given that $$\frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \equiv \frac { A } { ( 2 x + 1 ) } + \frac { B } { ( 2 x + 1 ) ^ { 2 } } + \frac { C } { ( x + 3 ) }$$

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find $$\int \frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \mathrm { d } x , \quad x > - \frac { 1 } { 2 }$$ (ii) Find $$\int \left( \mathrm { e } ^ { x } + 1 \right) ^ { 3 } \mathrm {~d} x$$ (iii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { 4 x + 5 x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x , \quad x > 0$$

Find \(\int \left( 12 x ^ { 5 } - 8 x ^ { 3 } + 3 \right) \mathrm { d } x\), giving each term in its simplest form.

A curve has equation \(y = \mathrm { f } ( x )\) and passes through the point (4, 22). Given that $$\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x ^ { \frac { 1 } { 2 } } - 7 ,$$ use integration to find \(\mathrm { f } ( x )\), giving each term in its simplest form.

Find $$\int \left( 12 x ^ { 5 } - 3 x ^ { 2 } + 4 x ^ { \frac { 1 } { 3 } } \right) \mathrm { d } x$$ giving each term in its simplest form.

Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),

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

Given that \(y = 3 x ^ { 2 } + 4 \sqrt { } x , x > 0\), find

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

Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form, (a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(b) \(\int y \mathrm {~d} x\).