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

1. Find

$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { 2 x ^ { 3 } } + 5 \right) d x$$ simplifying your answer.

12. The curve with equation \(y = \mathrm { f } ( x ) , x > 0\), passes through the point \(P ( 4 , - 2 )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x \sqrt { x } - 10 x ^ { - \frac { 1 } { 2 } }$$

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

1. Find, in simplest form,

$$\int \left( \frac { 8 x ^ { 3 } } { 3 } - \frac { 1 } { 2 \sqrt { x } } - 5 \right) \mathrm { d } x$$

11. A curve has equation \(y = \mathrm { f } ( x )\), where $$f ^ { \prime \prime } ( x ) = \frac { 6 } { \sqrt { x ^ { 3 } } } + x \quad x > 0$$ The point \(P ( 4 , - 50 )\) lies on the curve.
Given that \(\mathrm { f } ^ { \prime } ( x ) = - 4\) at \(P\),

1. find the equation of the normal at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
   (3)
2. find \(\mathrm { f } ( x )\).
   (8)

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9. (i) Find $$\int \frac { ( 3 x + 2 ) ^ { 2 } } { 4 \sqrt { x } } \mathrm {~d} x \quad x > 0$$ giving your answer in simplest form.
(ii) A curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given

• \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } + a x + b\) where \(a\) and \(b\) are constants
• the \(y\) intercept of \(C\) is - 8
• the point \(P ( 3 , - 2 )\) lies on \(C\)
• the gradient of \(C\) at \(P\) is 2
  find, in simplest form, \(\mathrm { f } ( x )\).
  

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1. Find

$$\int \left( \frac { 8 x ^ { 3 } } { 5 } - \frac { 2 } { 3 x ^ { 4 } } - 1 \right) d x$$ giving each term in simplest form.

6. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( x ) = \frac { ( x + 3 ) ^ { 2 } } { x \sqrt { x } }\)
• the point \(P ( 4,20 )\) lies on \(C\)
• Find \(\mathrm { f } ( x )\), simplifying your answer.
  

1. Find

$$\int \frac { 4 x ^ { 5 } + 3 } { 2 x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.

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

Given that

• \(\mathrm { f } ^ { \prime \prime } ( x ) = 4 x + \frac { 1 } { \sqrt { x } }\)
• the point \(P\) has \(x\) coordinate 4 and lies on \(C\)
• the tangent to \(C\) at \(P\) has equation \(y = 3 x + 4\)
  1. find an equation of the normal to \(C\) at \(P\)
  2. find \(\mathrm { f } ( x )\), writing your answer in simplest form.

1. Find

$$\int ( 2 x - 5 ) ( 3 x + 2 ) ( 2 x + 5 ) \mathrm { d } x$$ writing your answer in simplest form.

1. In this question you must show all stages of your working.

The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• the point \(P ( 2,8 \sqrt { 2 } )\) lies on \(C\)
• \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x ^ { 3 } } + \frac { k } { x ^ { 2 } }\) where \(k\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) at \(P\)
  1. find the exact value of \(k\),
  2. find \(\mathrm { f } ( x )\), giving your answer in simplest form.

1. Find

$$\int \frac { 4 x ^ { 2 } + 1 } { 2 \sqrt { x } } d x$$ giving the answer in its simplest form. $$\int \frac { 4 x ^ { 2 } + 1 } { 2 \sqrt { x } } \mathrm {~d} x$$ giving the answer in its simplest form.

1. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , \quad x > 0\), passes through the point \(P ( 4,1 )\).

Given that \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x } - 2 - \frac { 8 } { 3 x ^ { 2 } }\)

1. find the equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
   (4)
2. Find \(\mathrm { f } ( x )\).
   (5)
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4. Find $$\int \frac { ( 3 \sqrt { x } + 2 ) ( x - 5 ) } { 4 \sqrt { x } } d x$$ writing each term in simplest form.

6. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(C\) passes through the point \(P ( 8,2 )\)
• \(\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 3 x ^ { 2 } } + 3 - 2 ( \sqrt [ 3 ] { x } )\)
  1. find the equation of the tangent to \(C\) at \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
     (3)
  2. Find, in simplest form, \(\mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-21_2647_1840_118_111}

1. Find

$$\int \left( 10 x ^ { 5 } + 6 x ^ { 3 } - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x$$ giving your answer in simplest form.

7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \sqrt { x } } + \frac { A } { x ^ { 2 } } + 3\), where \(A\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) when \(x = 4\)
  1. find the value of \(A\).

Given also that

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Find the equation of the tangent to the curve with equation $$y = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$ at the point \(P ( 4,12 )\) Give your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers. The curve with equation \(y = \mathrm { f } ( x )\) also passes through the point \(P ( 4,12 )\) Given that $$f ^ { \prime } ( x ) = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$
2. find \(\mathrm { f } ( x )\) giving the coefficients in simplest form.

1. Find

$$\int \left( 10 x ^ { 4 } - \frac { 3 } { 2 x ^ { 3 } } - 7 \right) \mathrm { d } x$$ giving each term in simplest form.

9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 9,10 )\). Given that $$f ^ { \prime } ( x ) = 27 x ^ { 2 } - \frac { 21 x ^ { 3 } - 5 x } { 2 \sqrt { x } } \quad x > 0$$ find \(\mathrm { f } ( x )\), fully simplifying each term.

1. Find

$$\int 12 x ^ { 3 } + \frac { 1 } { 6 \sqrt { x } } - \frac { 3 } { 2 x ^ { 4 } } \mathrm {~d} x$$ giving each term in simplest form.

1. The curve \(C\) has equation

$$y = \frac { x ^ { 3 } } { 4 } - x ^ { 2 } + \frac { 17 } { x } \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(R \left( 2 , \frac { 13 } { 2 } \right)\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at the point \(R\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

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

Given that

• \(f ^ { \prime } ( x ) = \frac { 4 x ^ { 2 } + 10 - 7 x ^ { \frac { 1 } { 2 } } } { 4 x ^ { \frac { 1 } { 2 } } }\)
• the point \(P ( 4 , - 1 )\) lies on \(C\)
  1. (i) find the value of the gradient of \(C\) at \(P\) (ii) Hence find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
  2. Find \(\mathrm { f } ( x )\).

2. $$\mathrm { f } ( x ) = \frac { 8 } { x ^ { 2 } } - 4 \sqrt { x } + 3 x - 1 , \quad x > 0$$ Giving your answers in their simplest form, find

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

3. Find, using calculus and showing each step of your working, $$\int _ { 1 } ^ { 4 } \left( 6 x - 3 - \frac { 2 } { \sqrt { x } } \right) \mathrm { d } x$$