Indefinite & Definite Integrals

\(f'(x) = \left(2x - \frac{3}{x}\right)^2\) and \(f(3) = 2\) Find \(f(x)\). [4 marks]

9 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + 8 x\). The curve passes through the point \(( 1,5 )\). Find the equation of the curve.

It is given that $$\frac{\mathrm{d}y}{\mathrm{d}x} = (x + 2)(2x - 1)^2$$ and when \(x = 6\), \(y = 900\) Find \(y\) in terms of \(x\) [6 marks]

Find \(\int 7x^2 dx\). [3]

Simplify the following expressions fully.

1. \(\left( x ^ { 6 } \right) ^ { \frac { 1 } { 3 } }\)
2. \(\sqrt { 2 } \left( x ^ { 3 } \right) \div \sqrt { \frac { 32 } { x ^ { 2 } } }\)

5 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-06_355_634_255_753} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The point \(C\) lies on the circumference and angle \(A O C = \theta\) radians. The perimeter of sector \(B O C\) is twice the perimeter of sector \(A O C\). Find the value of \(\theta\) correct to 2 significant figures.

Find \(\int_2^5 (2x^3 + 3) dx\). [3]

2 It is given that $$\int _ { 0 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = 20 \text { and } \int _ { 3 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = - 10$$ Find the value of \(\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\) Circle your answer. \(- 30 - 101030\)

1. The graph of \(y = \mathrm { f } ( x )\) is shown below for \(0 \leq x \leq 6\) \includegraphics[max width=\textwidth, alt={}, center]{a1b449df-1096-4b3a-8306-fca410a7e530-04_499_551_331_877}

1. Evaluate \(\int _ { 0 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x\) [0pt] [2 marks]
2. Deduce values for each of the following, giving reasons for your answers.
   (b) (i) \(\int _ { 1 } ^ { 7 } \mathrm { f } ( x - 1 ) \mathrm { d } x\) (b) (ii) \(\int _ { 0 } ^ { 6 } ( \mathrm { f } ( x ) - 1 ) \mathrm { d } x\)

2 Find the value of the positive constant \(k\) for which \(\int _ { 1 } ^ { k } ( 2 x - 1 ) \mathrm { d } x = 6\).

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 \(k ^ { 3 } - 10 k ^ { 2 } - 7 k - 6 = 0\)

1. (a) Use the trapezium rule with two intervals of equal width to find an approximate value for the integral

$$\int _ { 0 } ^ { 2 } \arctan x \mathrm {~d} x$$ (b) Use the trapezium rule with four intervals of equal width to find an improved approximation for the value of the integral.

5 \includegraphics[max width=\textwidth, alt={}, center]{34177829-f05d-449e-8881-5ab4d852c4ce-3_458_643_285_751} The diagram shows the part of the curve \(y = x \mathrm { e } ^ { - x }\) for \(0 \leqslant x \leqslant 2\), and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { 2 } x \mathrm { e } ^ { - x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
3. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).

\includegraphics{figure_1} Figure 1 shows the graph of the curve with equation $$y = xe^x, \quad x \geq 0.$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.

1. Use integration to find the exact value of the area for \(R\). [5]
2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and \(0.8\).

   \(x\)	0	0.2	0.4	0.6	0.8
   \(y = xe^x\)	0	0.29836		1.99207

   [1]
3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures. [4]

In this question you must show detailed reasoning. Show that \(\int_5^{\infty} (x-1)^{-2} dx = 1\). [5]

1 Which of the integrals below is not an improper integral?
Circle your answer. \(\int _ { 0 } ^ { \infty } e ^ { - x } d x\) \(\int _ { 0 } ^ { 2 } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \sqrt { x } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\)

6

1. Find \(\int \left( x ^ { \frac { 1 } { 2 } } + 4 \right) \mathrm { d } x\).
2. 1. Find the value, in terms of \(a\), of \(\int _ { 1 } ^ { a } 4 x ^ { - 2 } \mathrm {~d} x\), where \(a\) is a constant greater than 1 .
   2. Deduce the value of \(\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x\).

3

1. Find \(\int ( 2 x + 1 ) ( x + 3 ) \mathrm { d } x\).
2. Evaluate \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).

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 \(x\).

Given that \(\text{f}(x) = (2x^{\frac{1}{3}} - 3x^{-\frac{1}{2}})^2 + 5\), \(x > 0\),

1. find, to 3 significant figures, the value of x for which f(x) = 5. [3]
2. Show that f(x) may be written in the form \(Ax^{\frac{2}{3}} + \frac{B}{x} + C\), where A, B and C are constants to be found. [3]
3. Hence evaluate \(\int_1^2 \text{f}(x) \, \text{dx}\). [5]

Show that \(\int_0^9 (3 + 4\sqrt{x})dx = 99\). [4]

1 In this question you must show detailed reasoning. Show that \(\int _ { 4 } ^ { 9 } ( 2 x + \sqrt { x } ) \mathrm { d } x = \frac { 233 } { 3 }\).

4

1. Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to \(\int _ { 0.5 } ^ { 2 } \frac { x } { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer to three significant figures.
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x ^ { 3 } } \mathrm {~d} x\).

1 Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).

The equation of a curve is such that \(\frac{dy}{dx} = \frac{1}{2}x + \frac{72}{x^4}\). The curve passes through the point \(P(2, 8)\).

1. Find the equation of the normal to the curve at \(P\). [2]
2. Find the equation of the curve. [4]

Find the exact value of \(\int_0^1 (e^x - x) dx\). [4]

14

1. Express \(\lim _ { \delta y \rightarrow 0 } \sum _ { 0 } ^ { h } \left( h ^ { 2 } - y ^ { 2 } \right) \delta y\) as an integral.
2. Hence show that \(V = \frac { 2 } { 3 } \pi r ^ { 2 } h\), as given in line 41 .

\includegraphics{figure_10} The diagram shows the graph of \(y = 1 - 3x^{-\frac{1}{2}}\).

1. Verify that the curve intersects the \(x\)-axis at \((9, 0)\). [1]
2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\)). Given that the area of the shaded region is 4 square units, find the value of \(a\). [9]

\includegraphics{figure_10} The diagram shows the curve \(y = \sin 2x \cos^2 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\).

1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis. [5]
2. Find the exact \(x\)-coordinate of \(M\). [6]

4. (a) Use the trapezium rule with two intervals of equal width to find an estimate for the value of the integral $$\int _ { 0 } ^ { 3 } e ^ { \cos x } d x$$ giving your answer to 3 significant figures.
(b) Use the trapezium rule with four intervals of equal width to find another estimate for the value of the integral to 3 significant figures.
(c) Given that the true value of the integral lies between the estimates made in parts (a) and (b), comment on the shape of the curve \(y = \mathrm { e } ^ { \cos x }\) in the interval \(0 \leq x \leq 3\) and explain your answer.
4. continued

6 \includegraphics[max width=\textwidth, alt={}, center]{52b43f20-e0e6-4ddd-9518-bea9782982bf-3_623_1354_262_392} The diagram shows the curve with equation \(y = 3 ^ { x }\) for \(0 \leqslant x \leqslant 1\). The area \(A\) under the curve between these limits is divided into \(n\) strips, each of width \(h\) where \(n h = 1\).

1. By using the set of rectangles indicated on the diagram, show that \(A > \frac { 2 h } { 3 ^ { h } - 1 }\).
2. By considering another set of rectangles, show that \(A < \frac { ( 2 h ) 3 ^ { h } } { 3 ^ { h } - 1 }\).
3. Given that \(h = 0.001\), use these inequalities to find values between which \(A\) lies.

Find the exact value of \(\int_0^{\frac{1}{4}\pi} (1 + \cos \frac{1}{2}x) dx\). [3]

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = 1 - 2 \cos x\), where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(A\) and at the point \(B\).

1. Find, in terms of \(\pi\), the \(x\) coordinate of the point \(A\) and the \(x\) coordinate of the point \(B\). The finite region \(S\) enclosed by the curve and the \(x\)-axis is shown shaded in Figure 3. The region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find, by integration, the exact value of the volume of the solid generated.

The function \(f\) is defined by \(f(x) = \frac{4x^2 + 12x + 9}{2x^2 + x - 3}\), where \(x > 1\).

1. Show that \(f(x)\) can be written as \(2 + \frac{5}{x-1}\). [3]
2. Hence find the exact value of \(\int_3^7 f(x)\,dx\). [4]

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-3_648_684_342_731}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

5 A particle \(P\) moves in a straight line. \(P\) starts from rest at \(O\) and travels to \(A\) where it comes to rest, taking 50 seconds. The speed of \(P\) at time \(t\) seconds after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v\) is defined as follows. $$\begin{aligned} \text { For } 0 \leqslant t \leqslant 5 , & v = t - 0.1 t ^ { 2 } \\ \text { for } 5 \leqslant t \leqslant 45 , & v \text { is constant } \\ \text { for } 45 \leqslant t \leqslant 50 , & v = 9 t - 0.1 t ^ { 2 } - 200 \end{aligned}$$

1. Find the distance travelled by \(P\) in the first 5 seconds.
2. Find the total distance from \(O\) to \(A\), and deduce the average speed of \(P\) for the whole journey from \(O\) to \(A\).