Integrate after algebraic manipulation

1 Find \(\int \left( x + \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\).

3 Show that \(\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } + 2 \mathrm { e } - \frac { 3 } { 2 }\).

6

1. Find
   1. \(\int \frac { \mathrm { e } ^ { 2 x } + 6 } { \mathrm { e } ^ { 2 x } } \mathrm {~d} x\),
   2. \(\int 3 \cos ^ { 2 } x \mathrm {~d} x\).
2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 1 } ^ { 2 } \frac { 6 } { \ln ( x + 2 ) } \mathrm { d } x$$ giving your answer correct to 2 decimal places.
   1. Express \(3 \cos \theta + \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
   2. Hence solve the equation $$3 \cos 2 x + \sin 2 x = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

2 Find the exact value of \(\int _ { 1 } ^ { 2 } \left( 2 \mathrm { e } ^ { 2 x } - 1 \right) ^ { 2 } \mathrm {~d} x\). Show all necessary working.

1. Find

$$\int \frac { 4 x ^ { 5 } + 3 } { 2 x ^ { 2 } } \mathrm {~d} x$$ giving 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. 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.

4. Find $$\int \frac { ( 3 \sqrt { x } + 2 ) ( x - 5 ) } { 4 \sqrt { x } } d x$$ writing each term in simplest form.

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.

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.

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 \(\int ( 4 + 3 \sqrt { } x ) ^ { 2 } \mathrm {~d} x\).

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 \(\mathrm { f } ( x )\).

1. Find

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

1. Find, in simplest form,

$$\int ( 2 \cos x - \sin x ) ^ { 2 } d x$$

1. The curve \(C\) has equation

$$y = 2 + \ln \left( 1 - x ^ { 2 } \right) \quad \frac { 1 } { 2 } \leqslant x \leqslant \frac { 3 } { 4 }$$

1. Show that the length of the curve \(C\) is given by $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x$$
2. Hence, using algebraic integration, show that the length of the curve \(C\) is \(p + \ln q\) where \(p\) and \(q\) are rational numbers to be determined.

2. Given that $$y = 2 x ^ { \frac { 3 } { 2 } } - 1 ,$$ find $$\int y ^ { 2 } \mathrm {~d} x .$$

2

1. Expand \(\left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 }\).
2. Hence find \(\int \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 } \mathrm {~d} x\).

1. Find

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

1. Find

$$\int \frac { 3 x ^ { 4 } - 4 } { 2 x ^ { 3 } } d x$$ writing your answer in simplest form.

1. Find

$$\int \frac { x ^ { \frac { 1 } { 2 } } ( 2 x - 5 ) } { 3 } \mathrm {~d} x$$ writing each term in simplest form.

7

1. Find \(\int x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x\).
2. Show that \(\int _ { 0 } ^ { 8 } x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x = a \times 2 ^ { 11 }\), where \(a\) is a positive integer to be determined.

2

1. Given that \(\frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }\) can be expressed as \(A x + \frac { B ( 4 x - 1 ) } { 2 x ^ { 2 } - x + 2 }\), find the values of the constants \(A\) and \(B\).
2. The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }$$ The point \(( - 1,2 )\) lies on the curve. Find the equation of the curve.
   [0pt] [4 marks]