# Integration - Past Edexcel Exam Questions

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1 Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( ) x 1 x dx. [4] 2. Question 2b - January The gradient of the curve C is given by The point P (1, 4) lies on C. dy dx = (3x 1)2 (a) Find an equation of the normal to C at P. [4] (b) Find an equation for the curve C in the form y = f(x). [5] (c) Using dy dx = (3x 1)2, show that there is no point on C at which the tangent is parallel to the line y = 1 2x. [2] Question 9 - January Given that y = 6x 4, x 0, x 2 (a) - (b) find y dx. [3] Question 2b - May (a) Show that (3 x) 2 x can be written as 9x x 1 2. [2] Given that dy = (3 x) 2 dx x, x > 0, and that y = 2 at x = 1, 3

2 (b) Find y in terms of x. [6] Question 7 - May Given that y = 2x 2 6, x 0, x 3 (a) - (b) find y dx. [3] Question 4b - January The curve with equation y = f(x) passes through the point (1,6). Given that f (x) = 3 + 5x2 + 2, x > 0, x 1 2 find f(x) and simplify your answer. [7]. Question 8 - January Find ( ) 6x x 1 2 dx, giving each term in its simplest form. [4]. Question 1 - May The curve C with equation y = f(x), x 0, passes through the point ( 3, 7 1 2). Given that f (x) = 2x + 3 x 2, (a) find f(x). [5] (b) Verify that f( 2) = 5. [1] (c) Find an equation for the tangent to C at the point (-2,5), giving your answer in the form ax + by + c = 0, where a, b and c are integers. [5] Question 10 - May 2006

3 9. (a) Show that (4 + 3 x) 2 can be written as 16 + k x + 9x, where k is a constant to be found. [2] (b) Find (4 + 3 x) 2 dx. [3] Question 6 - January The curve C has equation y = f(x), x 0, and the point P (2, 1) lies on C. Given that f (x) = 3x x 2, (a) find f(x). [5] (b) Find an equation for the tangent to C at the point P, giving your answer in the form y = mx + c, where m and c are integers. [4] Question 7 - January Given that y = 3x x, x > 0, find (a) - (b) - (c) y dx. [3] Question 3c - May The curve C with equation y = f(x) passes through the point (5,65). Given that f (x) = 6x 2 10x 12, (a) Use integration to find f(x). [4] (b) Hence show that f(x) = x(2x + 3)(x 4). [2] (c) Sketch C, showing the coordinates of the points where C crosses the x-axis. [3] Question 9 - May 2007

4 13. Find (3x 2 + 4x 5 7) dx. [4]. Question 1 - January The curve C has equation y = f(x), x > 0, and f (x) = 4x 6 x + 8. x 2 Given that the point P (4, 1) lies on C, (a) find f(x) and simplify your answer. [6] (b) Find an equation of the normal to C at the point P (4, 1). [4] Question 9 - January Find (2 + 5x 2 ) dx. [3]. Question 1 - June The gradient of a curve C is given by dy dx = (x2 +3) 2 x 2, x 0. (a) Show that dy dx = x x 2. [2] The point (3,20) lies on C. (b) Find an equation for the curve C in the form y = f(x). [6] Question 11 - June Find (12x 5 8x 3 + 3) dx, giving each term in its simplest form. [4] Question 2 - January A curve has equation y = f(x) and passes through the point (4, 22). Given that f (x) = 3x 2 3x 1 2 7,

5 use integration to find f(x), giving each term in its simplest form. [5]. Question 4 - January Given that y = 2x 3 + 3, x 0, find x 2 (a) - (b) y dx, simplifying each term. [3] Question 3b - June dy dx = 1 5x 2 + x x, x > 0 Given that y = 35 at x = 4, find y in terms of x, giving each term in its simplest form.. [7]. Question 4 - January Find ( 8x 3 + 6x ) dx, giving each term in its simplest form. [4]. Question 2 - May 2010

6 22. The curve C has equation y = f(x), x > 0, where dy dx = 3x 5 2. x Given that the point P (4, 5) lies on C, find (a) f(x), [5] (b) an equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0, where a, b and c are integers. [4] Question 11 - May Find ( ) 12x 5 3x 2 + 4x 1 3 dx, giving each term in its simplest form. [5]. Question 2 - January The curve with equation y = f(x) passes through the point (-1,0). Given that f (x) = 12x 2 8x + 1 find f(x). [5]. Question 7 - January Given that y = 2x , x 0, find, in their simplest form x 3 (a) - (b) y dx. [4] Question 2b - May Given that 6x+3x 5 2 x can be written in the form 6x p + 3xq,

7 (a) write down the value of p and the value of q. [2] Given that dy dx = 6x+3x 5 2 x and that y = 90 when x = 4, (b) find y in terms of x, simplifying the coefficient of each term. [5] Question 6 - May Given that y = x 4 + 6x 1 2, find in their simplest form (a) - (b) y dx. [3] Question 1b - January A curve with equation y = f(x) passes through the point (2,10). Given that f (x) = 3x 2 3x + 5, find the value of f(1). [5]. Question 7 - January Find (6x 2 + 2x ) dx giving each term in its simplest form. [4]. Question 1 - May The point P (4, 1) lies on the curve C with equation y = f(x), x > 0, and f (x) = 1 2 x 6 x + 3. (a) Find the equation of the tangent to C at the point P, giving your answer in the form y = mx + c, where m and c are integers. [4]

8 (b) Find f(x). [4] Question 7 - May dy dx = x3 + 4x 5, x 0 2x 3 Given that y = 7 at x = 1, find y in terms of x, giving each term in its simplest form.. [6]. Question 8 - January Find ( 10x 4 4x 3 x ) dx giving each term in its simplest form. [4]. Question 2 - May f (x) = (3 x2 ) 2, x 0 x 2 (a) Show that f (x) = 9x 2 + A + Bx 2, where A and B are constants to be found.[3] (b) Find f (x). [2] Given that the point (-3, 10) lies on the curve with equation y = f(x), (c) find f(x). [5] Question 9 - May Find (8x ) dx giving each term in its simplest form. [3]. Question 1 - May 2014

9 35. A curve with equation y = f(x) passes through the point (4, 25). Given that f (x) = 3 8 x2 10x , x > 0 (a) find f(x), simplifying each term. [5] (b) Find an equation of the normal to the curve at the point (4, 25). Give your answer in the form ax + by + c = 0, where a, b and c are integers to be found. [5] Question 10 - May 2014

10 Solutions 1. (a) - (b) x + 2x x 1 + c 2. (a) y = 1x (b) y = 3x 3 3x 2 + x + 3 (c) - 3. (a) - (b) 3x 2 + 4x 1 + c 4. (a) - (b) y = 18x 1 2 6x + 2x (a) - (b) 2 3 x3 + 3x 4 + c 6. y = 3x + 2x x x 3 + 2x + 2x c 8. (a) x 2 3x (b) - (c) 13x + 4y + 6 = 0 9. (a) k = 24 (b) 16x + 16x x2 + c 10. (a) x 3 6x + 8x (b) y = 4x (a) - (b) - (c) x 3 + 8x c (a) 2x 3 5x 2 12x (b) -

11 (c). y y = x(2x + 3)(x 4) ( 3 2, 0) (0, 0) (4, 0) x 13. x x6 7x + c 14. (a) 2x 2 4x 3 2 8x (b) y = 2x x x3 + c 16. (a) - (b) y = 1 3 x3 + 6x 9x x 6 2x 4 + 3x + c 18. x 3 2x 3 2 7x (a) - (b) 1 2 x4 3x 1 + c x x x 4 + 4x x + c 22. (a) 3 2 x2 10x 1 2 2x + 9 (b) 15x 2y 50 = x 6 x 3 + 3x c 24. 4x 3 4x 2 + x (a) -

12 (b) 1 3 x6 + 7x 1 2 x 2 + c 26. (a) p = 1 2, q = 2 (b) 4x x (a) (b) 1 5 x5 + 4x c 29. 2x 3 2x 1 + 5x + c 30. (a) y = 2x 9 (b) 1 4 x2 12x x x4 2x x x 5 2x 2 6x c 33. (a) A = 6, B = 1 (b) 18x 3 + 2x (c) 9x 1 6x x x 4 + 4x + c 35. (a) 1 8 x3 20x x + 53 (b) x + 2y 54 = 0

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