Indefinite & Definite Integrals

8 The gradient of a curve is \(3 \sqrt { x } - 5\). The curve passes through the point ( 4,6 ). Find the equation of the curve.

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.

1. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 1,6 )\). Given that

$$f ^ { \prime } ( x ) = 3 + \frac { 5 x ^ { 2 } + 2 } { x ^ { \frac { 1 } { 2 } } } , x > 0$$ find \(\mathrm { f } ( x )\) and simplify your answer.

4 Find \(\int _ { 2 } ^ { 5 } \left( 2 x ^ { 3 } + 3 \right) \mathrm { d } x\).

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.
3. 1. \(\int _ { 1 } ^ { 7 } \mathrm { f } ( x - 1 ) \mathrm { d } x\)
4. (ii) \(\int _ { 0 } ^ { 6 } ( \mathrm { f } ( x ) - 1 ) \mathrm { d } x\)

Find \(\int \left( 3 x ^ { 2 } + 4 x ^ { 5 } - 7 \right) d x\).

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.

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

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

2

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 1 } ^ { 5 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
2. 1. Find \(\int \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving the coefficient of each term in its simplest form.
   2. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\).

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).

6

1. 1. Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }\).
   2. By using an appropriate trigonometrical identity, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 2 } x \mathrm {~d} x\).
2. 1. Use the trapezium rule with 2 intervals to estimate the value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec x d x\), giving your answer correct to 2 significant figures.
   2. Determine, by sketching the appropriate part of the graph of \(y = \sec x\), whether the trapezium rule gives an under-estimate or an over-estimate of the true value.

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

2 Given that \(\mathrm { f } ( x ) = 3 x - 1\) find the mean value of \(\mathrm { f } ( x )\) over the interval \(4 \leq x \leq 8\) Circle your answer. 6111717

9 In an electrical circuit, the alternating current \(I\) amps is given by \(\mathbf { I } =\) asinnt, where \(t\) is the time in seconds and \(a\) and \(n\) are positive constants. The RMS value of the current, in amps, is defined to be the square root of the mean value of \(I ^ { 2 }\) over one complete period of \(\frac { 2 \pi } { n }\) seconds. Show that the RMS value of the current is \(\frac { a } { \sqrt { 2 } }\) amps.

6 Show that \(\int _ { 0 } ^ { 9 } ( 3 + 4 \sqrt { x } ) \mathrm { d } x = 99\).

6 Show that \(\int _ { 0 } ^ { 9 } ( 3 + 4 \sqrt { x } ) \mathrm { d } x = 99\).

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

6. Evaluate

1. \(\quad \int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\),
2. \(\int _ { - \infty } ^ { - 1 } \frac { 1 } { x ^ { 4 } } \mathrm {~d} x\).

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 \(\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x\).

7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x \neq 0\), and the point \(P ( 2,1 )\) lies on \(C\). Given that $$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 6 - \frac { 8 } { x ^ { 2 } } ,$$

1. find \(\mathrm { f } ( x )\).
2. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.

3

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

4. (i) Use Simpson's rule with four intervals, each of width 0.25 , to estimate the value of the integral $$\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ (ii) Find the exact value of the integral $$\int _ { \frac { 1 } { 2 } } ^ { 1 } e ^ { 1 - 2 x } d x$$

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 .

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.

6 The curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-5_428_563_372_740} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and cuts the \(y\)-axis at the point \(B\).

1. 1. State the coordinates of the point \(B\) and hence find the area of the triangle \(A O B\), where \(O\) is the origin.
   2. Find \(\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve and the line \(A B\).
2. 1. Find the gradient of the curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) at the point \(A ( - 1,0 )\).
   2. Hence find an equation of the tangent to the curve at the point \(A\).

9. \begin{figure}[h] \end{figure}

1. By using the substitution \(u = 2 x + 3\), show that $$\int _ { 0 } ^ { 12 } \frac { x } { ( 2 x + 3 ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3 - \frac { 2 } { 9 }$$ The curve \(C\) has equation $$y = \frac { 9 \sqrt { x } } { ( 2 x + 3 ) } , \quad x > 0$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 12\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Use the result of part (a) to find the exact value of the volume of the solid generated.

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 \(A\).

4

1. Express \(\frac { 1 } { ( x - 1 ) ( x + 2 ) }\) in partial fractions
2. In this question you must show detailed reasoning. Hence find \(\int _ { 2 } ^ { 3 } \frac { 1 } { ( x - 1 ) ( x + 2 ) } \mathrm { d } x\).
   Give your answer in its simplest form.

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.