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1EDEXCEL C4 June 2012 Q7
Figure 3 shows a sketch of part of the curve with equation y = x^(1/2) ln 2x. The finite region R, shown shaded in Figure 3, is bounded by the curve, the x-axis and the lines x = 1 and x = 4. (a) Use the trapezium rule, with 3 strips of equal width, to find an estimate for the area of R, giving your answer to 2 decimal places. (4) (b) Find the integral of x^(1/2) ln 2x dx. (4) (c) Hence find the exact area of R, giving your answer in the form a ln 2 + b, where a and b are exact constants. (3)
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2EDEXCEL C4 January 2012 Q2
(a) Use integration by parts to find the integral of x sin 3x dx. (3) (b) Using your answer to part (a), find the integral of x^2 cos 3x dx. (3)
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3EDEXCEL C4 January 2012 Q6
Figure 3 shows a sketch of the curve with equation y = 2 sin 2x / (1 + cos x), 0 <= x <= pi/2. The finite region R, shown shaded in Figure 3, is bounded by the curve and the x-axis. The table below shows corresponding values of x and y for y = 2 sin 2x / (1 + cos x): x = 0, pi/8, pi/4, 3pi/8, pi/2 with y = 0, blank, 1.17157, 1.02280, 0. (a) Complete the table above giving the missing value of y to 5 decimal places. (1) (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 4 decimal places. (3) (c) Using the substitution u = 1 + cos x, or otherwise, show that the integral of 2 sin 2x / (1 + cos x) dx = 4 ln(1 + cos x) - 4 cos x + k, where k is a constant. (5) (d) Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures. (3)
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4EDEXCEL C4 June 2011 Q4
Figure 2 shows a sketch of the curve with equation y = x^3 ln(x^2 + 2), x >= 0. The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the line x = sqrt(2). The table below shows corresponding values of x and y for y = x^3 ln(x^2 + 2): x = 0, sqrt(2)/4, sqrt(2)/2, 3 sqrt(2)/4, sqrt(2) with y = 0, blank, 0.3240, blank, 3.9210. (a) Complete the table above giving the missing values of y to 4 decimal places. (2) (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places. (3) (c) Use the substitution u = x^2 + 2 to show that the area of R is (1/2) times the integral from 2 to 4 of (u - 2) ln u du. (4) (d) Hence, or otherwise, find the exact area of R. (6)
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5EDEXCEL C4 June 2011 Q8
(a) Find the integral of (4y + 3)^(-1/2) dy. (2) (b) Given that y = 1.5 at x = -2, solve the differential equation dy/dx = sqrt(4y + 3) / x^2, giving your answer in the form y = f(x). (6)
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6EDEXCEL C4 January 2011 Q1
Use integration to find the exact value of the integral from 0 to pi/2 of x sin 2x dx. (6)
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7EDEXCEL C4 January 2011 Q3
(a) Express 5 / ((x - 1)(3x + 2)) in partial fractions. (3) (b) Hence find the integral of 5 / ((x - 1)(3x + 2)) dx, where x > 1. (3) (c) Find the particular solution of the differential equation (x - 1)(3x + 2) dy/dx = 5y, x > 1, for which y = 8 at x = 2. Give your answer in the form y = f(x). (6)
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8EDEXCEL C4 January 2011 Q7
I = integral from 2 to 5 of 1 / (4 + sqrt(x - 1)) dx. (a) Given that y = 1 / (4 + sqrt(x - 1)), complete the table below with values of y corresponding to x = 3 and x = 5. Give your values to 4 decimal places: x = 2, 3, 4, 5 with y = 0.2, blank, 0.1745, blank. (2) (b) Use the trapezium rule, with all of the values of y in the completed table, to obtain an estimate of I, giving your answer to 3 decimal places. (4) (c) Using the substitution x = (u - 4)^2 + 1, or otherwise, and integrating, find the exact value of I. (8)
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9EDEXCEL C4 June 2010 Q2
Using the substitution u = cos x + 1, or otherwise, show that the integral from 0 to pi/2 of e^(cos x + 1) sin x dx = e(e - 1). (6)
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10EDEXCEL C4 June 2010 Q6
f(theta) = 4 cos^2 theta - 3 sin^2 theta. (a) Show that f(theta) = 1/2 + (7/2) cos 2theta. (3) (b) Hence, using calculus, find the exact value of the integral from 0 to pi/2 of theta f(theta) dtheta. (7)
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11EDEXCEL C4 January 2010 Q2
Figure 1 shows a sketch of the curve with equation y = x ln x, x >= 1. The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 4. The table shows corresponding values of x and y for y = x ln x: x = 1, 1.5, 2, 2.5, 3, 3.5, 4 with y = 0, 0.608, blank, blank, 3.296, 4.385, 5.545. (a) Complete the table with the values of y corresponding to x = 2 and x = 2.5, giving your answers to 3 decimal places. (2) (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places. (4) (c) (i) Use integration by parts to find the integral of x ln x dx. (ii) Hence find the exact area of R, giving your answer in the form (1/4)(a ln 2 + b), where a and b are integers. (7)
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12EDEXCEL C4 January 2010 Q5
(a) Find the integral of (9x + 6)/x dx, x > 0. (2) (b) Given that y = 8 at x = 1, solve the differential equation dy/dx = ((9x + 6) y^(1/3)) / x, giving your answer in the form y^2 = g(x). (6)
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13EDEXCEL C4 January 2010 Q8
(a) Using the substitution x = 2 cos u, or otherwise, find the exact value of the integral from 1 to sqrt(2) of 1 / (x^2 sqrt(4 - x^2)) dx. (7) Figure 3 shows a sketch of part of the curve with equation y = 4 / (x (4 - x^2)^(1/4)), 0 < x < 2. The shaded region S, shown in Figure 3, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = sqrt(2). The shaded region S is rotated through 2pi radians about the x-axis to form a solid of revolution. (b) Using your answer to part (a), find the exact volume of the solid of revolution formed. (3)
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14EDEXCEL C4 June 2009 Q3
f(x) = (4 - 2x) / ((2x + 1)(x + 1)(x + 3)) = A/(2x + 1) + B/(x + 1) + C/(x + 3). (a) Find the values of the constants A, B and C. (4) (b) (i) Hence find the integral of f(x) dx. (3) (ii) Find the integral from 0 to 2 of f(x) dx in the form ln k, where k is a constant. (3)
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15EDEXCEL C4 June 2009 Q6
(a) Find the integral of sqrt(5 - x) dx. (2) Figure 3 shows a sketch of the curve with equation y = (x - 1) sqrt(5 - x), 1 <= x <= 5. (b) (i) Using integration by parts, or otherwise, find the integral of (x - 1) sqrt(5 - x) dx. (4) (ii) Hence find the integral from 1 to 5 of (x - 1) sqrt(5 - x) dx. (2)
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16EDEXCEL C4 January 2009 Q6
(a) Find the integral of tan^2 x dx. (2) (b) Use integration by parts to find the integral of (1/x^3) ln x dx. (4) (c) Use the substitution u = 1 + e^x to show that the integral of e^(3x) / (1 + e^x) dx = (1/2) e^(2x) - e^x + ln(1 + e^x) + k, where k is a constant. (7)
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17EDEXCEL C4 June 2008 Q2
(a) Use integration by parts to find the integral of x e^x dx. (3) (b) Hence find the integral of x^2 e^x dx. (3)
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18EDEXCEL C4 June 2008 Q8
Figure 3 shows the curve C with parametric equations x = 8 cos t, y = 4 sin 2t, 0 <= t <= pi/2. The point P lies on C and has coordinates (4, 2 sqrt(3)). (a) Find the value of t at the point P. (2) The line l is a normal to C at P. (b) Show that an equation for l is y = -x sqrt(3) + 6 sqrt(3). (6) The finite region R is enclosed by the curve C, the x-axis and the line x = 4, as shown shaded in Figure 3. (c) Show that the area of R is given by the integral from pi/3 to pi/2 of 64 sin^2 t cos t dt. (4) (d) Use this integral to find the area of R, giving your answer in the form a + b sqrt(3), where a and b are constants to be determined. (4)
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19EDEXCEL C4 January 2008 Q4
(i) Find the integral of ln(x/2) dx. (4) (ii) Find the exact value of the integral from pi/4 to pi/2 of sin^2 x dx. (5)
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20Edexcel A Level June 2023 Paper 2 Q5
The curve C has equation y = f(x). The curve passes through the point P (3, -10) and has a turning point at P. Given that dy/dx = 2x^3 - 9x^2 + 5x + k, where k is a constant, (a) show that k = 12 (2) (b) Hence find the coordinates of the point where C crosses the y-axis. (3)
21EDEXCEL C4 January 2008 Q7
Figure 3. The curve C has parametric equations x = ln(t + 2), y = 1/(t + 1), t > -1. The finite region R between the curve C and the x-axis, bounded by the lines with equations x = ln 2 and x = ln 4, is shown shaded in Figure 3. (a) Show that the area of R is given by the integral from 0 to 2 of 1 / ((t + 1)(t + 2)) dt. (4) (b) Hence find an exact value for this area. (6) (c) Find a cartesian equation of the curve C, in the form y = f(x). (4) (d) State the domain of values for x for this curve. (1)
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22EDEXCEL C4 January 2008 Q8
Liquid is pouring into a large vertical circular cylinder at a constant rate of 1600 cm^3/s and is leaking out of a hole in the base, at a rate proportional to the square root of the height of the liquid already in the cylinder. The area of the circular cross section of the cylinder is 4000 cm^2. (a) Show that at time t seconds, the height h cm of liquid in the cylinder satisfies the differential equation dh/dt = 0.4 - k sqrt(h), where k is a positive constant. (3) When h = 25, water is leaking out of the hole at 400 cm^3/s. (b) Show that k = 0.02. (1) (c) Separate the variables of the differential equation dh/dt = 0.4 - 0.02 sqrt(h), to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by the integral from 0 to 100 of 50 / (20 - sqrt(h)) dh. (2) Using the substitution h = (20 - x)^2, or otherwise, (d) find the exact value of the integral from 0 to 100 of 50 / (20 - sqrt(h)) dh. (6) (e) Hence find the time taken to fill the cylinder from empty to a height of 100 cm, giving your answer in minutes and seconds to the nearest second. (1)
