Integration with given constant

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 \(k\).

8. Given \(k > 3\) and $$\int _ { 3 } ^ { k } \left( 2 x + \frac { 6 } { x ^ { 2 } } \right) \mathrm { d } x = 10 k$$ show that \(k ^ { 3 } - 10 k ^ { 2 } - 7 k - 6 = 0\)

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 \(b\).

9. (i) Evaluate $$\int _ { 1 } ^ { 3 } ( 3 - \sqrt { x } ) ^ { 2 } d x$$ giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
(ii) The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 4 x + k$$ where \(k\) is a constant. Given that the curve passes through the points \(( 0 , - 2 )\) and \(( 2,18 )\), show that \(k = 2\) and find an equation for the curve.

9 The positive constant \(a\) is such that \(\int _ { a } ^ { 2 a } \frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 4 } { x ^ { 2 } } \mathrm {~d} x = 0\).

1. Show that \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\).
2. Show that \(a = 1\) is a root of \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\), and hence find the other possible value of \(a\), giving your answer in simplified surd form.

1. Find the value of the constant \(k , 0 < k < 9\), such that

$$\int _ { k } ^ { 9 } \frac { 6 } { \sqrt { x } } \mathrm {~d} x = 20$$

1. Given that \(A\) is constant and

$$\int _ { 1 } ^ { 4 } ( 3 \sqrt { x } + A ) \mathrm { d } x = 2 A ^ { 2 }$$ show that there are exactly two possible values for \(A\).

1

1. Find \(\int \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x\), where \(a\) is a constant.
2. Hence, given that \(\int _ { 1 } ^ { 3 } \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x = 16\), find the value of the constant \(a\).
   [0pt] [2 marks]

1. Given that

$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).

8 In this question you must show detailed reasoning. Given that \(\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7\), find the value of \(a\).

3

1. Amaya and Ben integrated \(( 1 + x ) ^ { 2 }\), with respect to \(x\), using different methods, as follows. Amaya: \(\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \frac { ( 1 + x ) ^ { 3 } } { 3 } + c \quad = \frac { 1 } { 3 } + x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c\) Ben: \(\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \int \left( 1 + 2 x + x ^ { 2 } \right) \mathrm { d } x = x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c\) Charlie said that, because these answers are different, at least one of them must be wrong. Explain whether you agree with Charlie's statement.
2. You are given that \(a\) is a constant greater than 1 .
   1. Find \(\int _ { 1 } ^ { a } \frac { 1 } { ( 1 + x ) ^ { 2 } } \mathrm {~d} x\), giving your answer as a single fraction in terms of the constant \(a\).
   2. You are given that the area enclosed by the curve \(y = \frac { 1 } { ( 1 + x ) ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = a\) is equal to \(\frac { 1 } { 3 }\). Determine the value of \(a\).
3. In this question you must show detailed reasoning. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } \frac { \cos 2 x } { \sin 2 x + 2 } \mathrm {~d} x\), giving your answer in its simplest form.