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

7. (i) Find $$\int \frac { 2 + 4 x ^ { 3 } } { x ^ { 2 } } \mathrm {~d} x$$ giving each term in its simplest form.
(ii) Given that \(k\) is a constant and $$\int _ { 2 } ^ { 4 } \left( \frac { 4 } { \sqrt { x } } + k \right) \mathrm { d } x = 30$$ find the exact value of \(k\).

1. Given that

$$y = \frac { 2 x ^ { \frac { 2 } { 3 } } + 3 } { 6 } , \quad x > 0$$ find, in the simplest form,

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

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

$$f ^ { \prime } ( x ) = 3 \sqrt { x } - \frac { 9 } { \sqrt { x } } + 2$$ Given that the point \(P ( 9,14 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying your answer,
2. find an equation of the normal 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. (a) Show that \(\frac { x ^ { 2 } - 4 } { 2 \sqrt { } x }\) can be written in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants to be determined.
(b) Hence find $$\int \frac { x ^ { 2 } - 4 } { 2 \sqrt { x } } \mathrm {~d} x , \quad x > 0$$ giving your answer in its simplest form.

8. (a) Find \(\int \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x\), simplifying each term. Given that \(b\) is a constant and $$\int _ { b } ^ { 4 } \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x = 36$$ (b) show that \(b ^ { 3 } + 2 b ^ { 2 } - 15 b = 0\) (c) Hence find the possible values of \(b\).

11. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$f ^ { \prime } ( x ) = \frac { 5 x ^ { 2 } + 4 } { 2 \sqrt { x } } - 5$$ It is given that the point \(P ( 4,14 )\) lies on \(C\).

1. Find \(\mathrm { f } ( x )\), writing each term in a simplified form.
2. 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 constants.

1. $$f ( x ) = 3 x ^ { 2 } + x - \frac { 4 } { \sqrt { x } } + 6 x ^ { - 3 } , \quad x > 0$$ Find \(\int \mathrm { f } ( x ) \mathrm { d } x\), simplifying each term.

3. (a) Express \(\frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } }\) in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants.
(b) Hence find $$\int \frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } } d x$$ simplifying your answer.

3. Given that \(y = 2 x ^ { 3 } - \frac { 5 } { 3 x ^ { 2 } } + 7 , x \neq 0\), find in its simplest form

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

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10. \begin{figure}[h] \end{figure} The finite region \(R\), which is shown shaded in Figure 1, is bounded by the coordinate axes, the straight line \(l\) with equation \(y = \frac { 1 } { 3 } x + 5\) and the curve \(C\) with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - x + 5 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(D\) on the \(y\)-axis and at the point \(E\), as shown in Figure 1.

1. Use algebra to find the coordinates of the points \(D\) and \(E\). The curve \(C\) crosses the \(x\)-axis at the point \(F\).
2. Verify that the \(x\) coordinate of \(F\) is 25
3. Use algebraic integration to find the exact area of the shaded region \(R\).

4. Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form,

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

4. Given that \(y = 2 x ^ { 2 } - \frac { 6 } { x ^ { 3 } } , x \neq 0\),

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

6. (a) Show that \(( 4 + 3 \sqrt { } x ) ^ { 2 }\) can be written as \(16 + k \sqrt { } x + 9 x\), where \(k\) is a constant to be found.
(b) Find \(\int ( 4 + 3 \sqrt { } x ) ^ { 2 } \mathrm {~d} x\).

Find \(\int \left( 3 x ^ { 2 } + 4 x ^ { 5 } - 7 \right) d x\).

4. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 35\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.

7. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( - 1,0 )\). Given that $$\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 8 x + 1$$ find \(\mathrm { f } ( x )\).

Given that \(y = x ^ { 4 } + 6 x ^ { \frac { 1 } { 2 } }\), find in their simplest form

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

1. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,10 )\). Given that

$$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x + 5$$ find the value of \(\mathrm { f } ( 1 )\).

8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 3 } + \frac { 4 x - 5 } { 2 x ^ { 3 } } , \quad x \neq 0$$ Given that \(y = 7\) at \(x = 1\), find \(y\) in terms of \(x\), giving each term in its simplest form.

9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point ( 3,6 ). Given that $$f ^ { \prime } ( x ) = ( x - 2 ) ( 3 x + 4 )$$

1. use integration to find \(\mathrm { f } ( x )\). Give your answer as a polynomial in its simplest form.
2. Show that \(\mathrm { f } ( x ) \equiv ( x - 2 ) ^ { 2 } ( x + p )\), where \(p\) is a positive constant. State the value of \(p\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any points where the curve touches or crosses the coordinate axes.

7. (a) Show that \(\frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x }\) can be written as \(9 x ^ { - \frac { 1 } { 2 } } - 6 + x ^ { \frac { 1 } { 2 } }\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , x > 0\), and that \(y = \frac { 2 } { 3 }\) at \(x = 1\),
(b) find \(y\) in terms of \(x\).

Find \(\int \left( 6 x ^ { 2 } + 2 + x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving each term in its simplest form.

10. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \neq 0\), passes through the point ( \(3,7 \frac { 1 } { 2 }\) ). Given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 3 } { x ^ { 2 } }\),

1. find \(\mathrm { f } ( x )\).
2. Verify that \(f ( - 2 ) = 5\).
3. Find an equation for the tangent to \(C\) at the point ( \(- 2,5\) ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

9. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 5,65 )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - 10 x - 12\),

1. use integration to find \(\mathrm { f } ( x )\).
2. Hence show that \(\mathrm { f } ( x ) = x ( 2 x + 3 ) ( x - 4 )\).
3. In the space provided on page 17, sketch \(C\), showing the coordinates of the points where \(C\) crosses the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-11_76_40_2646_1894}

Find \(\int \left( 2 + 5 x ^ { 2 } \right) d x\).