Independent multi-part (different techniques)

8

1. Show that \(\int _ { 2 } ^ { 4 } 4 x \ln x \mathrm {~d} x = 56 \ln 2 - 12\).
2. Use the substitution \(u = \sin 4 x\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 24 } \pi } \cos ^ { 3 } 4 x \mathrm {~d} x\).

1. (i) Find

$$\int \frac { 2 x ^ { 2 } + 5 x + 1 } { x ^ { 2 } } \mathrm {~d} x , \quad x > 0$$ (ii) Find $$\int x \cos 2 x \mathrm {~d} x$$

4. (i) Find \(\int \ln \left( \frac { x } { 2 } \right) \mathrm { d } x\).
(ii) Find the exact value of \(\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } x \mathrm {~d} x\).

6. (a) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
(b) Use integration by parts to find \(\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x\).
(c) Use the substitution \(u = 1 + e ^ { x }\) to show that $$\int \frac { \mathrm { e } ^ { 3 x } } { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - \mathrm { e } ^ { x } + \ln \left( 1 + \mathrm { e } ^ { x } \right) + k$$ where \(k\) is a constant.

2. (i) Find $$\int x \cos \left( \frac { x } { 2 } \right) \mathrm { d } x$$ (ii) (a) Express \(\frac { 1 } { x ^ { 2 } ( 1 - 3 x ) }\) in partial fractions.
(b) Hence find, for \(0 < x < \frac { 1 } { 3 }\) $$\int \frac { 1 } { x ^ { 2 } ( 1 - 3 x ) } \mathrm { d } x$$

1. (i) Find

$$\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$$ (4)
(ii) Use the substitution \(u = \sqrt { 1 - 3 x }\) to show that $$\int \frac { 27 x } { \sqrt { 1 - 3 x } } \mathrm {~d} x = - 2 ( 1 - 3 x ) ^ { \frac { 1 } { 2 } } ( A x + B ) + k$$ where \(A\) and \(B\) are integers to be found and \(k\) is an arbitrary constant.

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

\section*{Solutions based on calculator technology are not acceptable.}

1. Use integration by parts to find the exact value of $$\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ giving your answer in simplest form.
2. Use integration by substitution to show that $$\int _ { 3 } ^ { \frac { 21 } { 2 } } \frac { 4 x } { ( 2 x - 1 ) ^ { 2 } } \mathrm {~d} x = a + \ln b$$ where \(a\) and \(b\) are constants to be found.

6

1. Find \(\int x \cos 2 x d x\).
2. Using the substitution \(u = x ^ { 2 } + 1\), or otherwise, find the exact value of \(\int _ { 2 } ^ { 3 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x\).

3. Find

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

1. (i) Find

$$\int x ^ { 2 } \sin x \mathrm {~d} x$$ (ii) Use the substitution \(u = 1 + \sin x\) to find the value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos x ( 1 + \sin x ) ^ { 3 } d x$$

8. (i) Find $$\int x ^ { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x$$ (ii) Using the substitution \(u = \sin t\), evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$

4

1. Use integration by parts to find \(\int x \sin x \mathrm {~d} x\).
2. Using the substitution \(u = x ^ { 2 } + 5\), or otherwise, find \(\int x \sqrt { x ^ { 2 } + 5 } \mathrm {~d} x\).
3. The diagram shows the curve \(y = x ^ { 2 } - 9\) for \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-3_844_663_685_694} The shaded region \(R\) is bounded by the curve, the lines \(y = 1\) and \(y = 2\), and the \(y\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

5

1. Find \(\int \frac { 1 } { 3 + 2 x } \mathrm {~d} x\).
2. By using integration by parts, find \(\int x \sin \frac { x } { 2 } \mathrm {~d} x\).

3

1. Find \(\int \mathrm { e } ^ { 4 x } \mathrm {~d} x\).
2. Use integration by parts to find \(\int \mathrm { e } ^ { 4 x } ( 2 x + 1 ) \mathrm { d } x\).
3. By using the substitution \(u = 1 + \ln x\), or otherwise, find \(\int \frac { 1 + \ln x } { x } \mathrm {~d} x\).

8

1. Using integration by parts, find \(\int x \sin ( 2 x - 1 ) \mathrm { d } x\).
2. Use the substitution \(u = 2 x - 1\) to find \(\int \frac { x ^ { 2 } } { 2 x - 1 } \mathrm {~d} x\), giving your answer in terms of \(x\).
   (6 marks)

6

1. Use integration by parts to find \(\int x \mathrm { e } ^ { 5 x } \mathrm {~d} x\).
2. 1. Use the substitution \(u = \sqrt { x }\) to show that $$\int \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x = \int \frac { 2 } { 1 + u } \mathrm {~d} u$$
   2. Find the exact value of \(\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x\).

5

1. Find \(\int \left( \frac { 1 } { x - 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\) giving your answer in its simplest form.
2. Use integration by parts to find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).

8

1. Evaluate exactly \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - x } \mathrm {~d} x\).
2. Find \(\int \frac { x - 1 } { x + 1 } \mathrm {~d} x\).

1. Use integration by parts to show that $$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2}\ln 2.$$ [6]

\includegraphics{figure_1} The finite region \(R\), bounded by the equation \(y = x^{\frac{1}{2}} \sec x\), the line \(x = \frac{\pi}{4}\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Find the volume of the solid of revolution generated. [2]
2. Find the gradient of the curve with equation \(y = x^{\frac{1}{2}} \sec x\) at the point where \(x = \frac{\pi}{4}\). [3]

\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 + 1}{(1 + x)(3 - x)}, \quad 0 \leq x < 3.$$

1. Given that \(f(x) = A + \frac{B}{1 + x} + \frac{C}{3 - x}\), find the values of the constants \(A\), \(B\) and \(C\). [4]

The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).

1. Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p\), \(q\) and \(r\) are rational constants to be found. [5]

1. Use integration by parts to show that $$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2} \ln 2.$$ [6]

\includegraphics{figure_1} The finite region \(R\), bounded by the equation \(y = x^{\frac{1}{2}} \sec x\), the line \(x = \frac{\pi}{4}\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Find the volume of the solid of revolution generated. [2]
2. Find the gradient of the curve with equation \(y = x^{\frac{1}{2}} \sec x\) at the point where \(x = \frac{\pi}{4}\). [3]

Evaluate

1. \(\int_0^2 x^3 \ln x \, dx\). [6]
2. \(\int_0^1 \frac{2+x}{\sqrt{4-x^2}} \, dx\). [6]