C1 Integration - Questions.pdf for engineering students
- 1. Solomon Press INTEGRATION C1 Worksheet A 1 Integrate with respect to x a x2 b x6 c x d x−4 e 5 f 3x2 g 4x7 h 6x−2 i 8x5 j 1 3 x k 2x−9 l 3 4 x−3 2 Find a ∫ (2x + 3) dx b ∫ (12x3 − 4x) dx c ∫ (7 − x2 ) dx d ∫ (x2 + x + 1) dx e ∫ (x4 + 5x2 ) dx f ∫ x(x2 − 3) dx g ∫ (x − 2)2 dx h ∫ (3x4 + x2 − 6) dx i ∫ (2 + 2 1 x ) dx j ∫ (x − 3 1 x ) dx k ∫ x2 ( 4 2 x − 3) dx l ∫ (x − 4 x )2 dx 3 Integrate with respect to y a 1 2 y b 5 2 y c 1 2 y − d 1 3 4y e 3 4 y f 2 3 5y − g 4 y h 7 y i 2 1 2y j 3 y k 4 5 2y l 1 3 y 4 Find a ∫ ( 1 2 3t − 1) dt b ∫ (2r + r ) dr c ∫ (3p − 1)2 dp d ∫ (4x + 1 3 x ) dx e ∫ ( 3 1 y + y) dy f ∫ ( 1 2 x2 − 3 2 x ) dx g ∫ 3 2 t t t + dt h ∫ ( 5 3 r − 2 3 r ) dr i ∫ 4 2 4 2 p p p − dp j ∫ (4 − 7 4 y ) dy k ∫ 2 2 1 6 3 x x + dx l ∫ 2 3 t t + dt 5 Find ∫ y dx when a y = 3x2 − x + 6 b y = x6 − x3 + 2x − 5 c y = x(x − 2)(x + 1) d y = ( 1 2 x + 2)2 e y = (x2 − 4)(2x + 3) f y = x3 − 4 3 2x + 2 7 x g y = 3 1 4x − 2 2 3x h y = (1 − 2 2 x )2 i y = ( 5 2 x − 1)( 3 2 x + 1) 6 Find a general expression for y given that a d d y x = 8x + 3 b d d y x = 1 2 x3 − x2 c d d y x = 3 4 3x d d d y x = (x + 1)3 e d d y x = 2x − 3 x f d d y x = 3 2 x − 3 2 2x − g d d y x = 2 2 3 2 x x − h d d y x = 3 2 x (5 − x) i d d y x = 9 2 3 x x − PMT
- 2. Solomon Press INTEGRATION C1 Worksheet B 1 a Find ∫ (2x + 1) dx. b Given that d d y x = 2x + 1 and that y = 5 when x = 1, find an expression for y in terms of x. 2 Use the given boundary conditions to find an expression for y in each case. a d d y x = 3 − 6x, y = 1 at x = 2 b d d y x = 3x2 − x, y = 41 at x = 4 c d d y x = x2 + 4x + 1, y = 4 at x = −3 d d d y x = 7 − 5x − x3 , y = 0 at x = 2 e d d y x = 8x − 2 2 x , y = −1 at x = 1 2 f d d y x = 3 − x , y = 8 at x = 4 3 The curve y = f(x) passes through the point (3, 5). Given that f ′(x) = 3 + 2x − x2 , find an expression for f(x). 4 Given that d d y x = 3 2 10x − 1 2 2x − , and that y = 7 when x = 0, find the value of y when x = 4. 5 The curve y = f(x) passes through the point (−1, 4). Given that f ′(x) = 2x3 − x − 8, a find an expression for f(x), b find an equation of the tangent to the curve at the point on the curve with x-coordinate 2. 6 The curve y = f(x) passes through the origin. Given that f ′(x) = 3x2 − 8x − 5, find the coordinates of the other points where the curve crosses the x-axis. 7 Given that d d y x = 3x + 2 2 x , a find an expression for y in terms of x. Given also that y = 8 when x = 2, b find the value of y when x = 1 2 . 8 The curve C with equation y = f(x) is such that d d y x = 3x2 + kx, where k is a constant. Given that C passes through the points (1, 6) and (2, 1), a find the value of k, b find an equation of the curve. PMT
- 3. Solomon Press INTEGRATION C1 Worksheet C 1 Find ∫ (x2 + 6 x − 3) dx. (3) 2 The curve y = f(x) passes through the point (1, −2). Given that f ′(x) = 1 − 3 6 x , a find an expression for f(x). (4) The point A on the curve y = f(x) has x-coordinate 2. b Show that the normal to the curve y = f(x) at A has the equation 16x + 4y − 19 = 0. (5) 3 The curve y = f(x) passes through the point (3, 22). Given that f ′(x) = 3x2 + 2x − 5, a find an expression for f(x). (4) Given also that g(x) = (x + 3)(x − 1)2 , b show that g(x) = f(x) + 2, (3) c sketch the curves y = f(x) and y = g(x) on the same set of axes. (3) 4 Given that y = x2 − 2 3 x , find a d d y x , (2) b ∫ y dx. (3) 5 The curve C with equation y = f(x) is such that d d y x = 3x2 − 4x − 1. Given that the tangent to the curve at the point P with x-coordinate 2 passes through the origin, find an equation for the curve. (7) 6 A curve with equation y = f(x) is such that d d y x = 3 x − 2 x , x > 0. a Find the gradient of the curve at the point where x = 2, giving your answer in its simplest form. (2) Given also that the curve passes through the point (4, 7), b find the y-coordinate of the point on the curve where x = 3, giving your answer in the form 3 a + b, where a and b are integers. (6) PMT
- 4. Solomon Press 7 Find a ∫ (x + 2)2 dx, (3) b ∫ 1 4 x dx. (3) 8 The curve C has the equation y = f(x) and crosses the x-axis at the point P (−2, 0). Given that f ′(x) = 3x2 − 2x − 3, a find an expression for f(x), (4) b show that the tangent to the curve at the point where x = 1 has the equation y = 5 − 2x. (3) 9 Given that d d y x = 2x − 2 3 x , x ≠ 0, and that y = 0 at x = 1, a find an expression for y in terms of x, (4) b show that for all non-zero values of x x2 2 2 d d y x − 2y = k, where k is a constant to be found. (4) 10 Integrate with respect to x a 3 1 x , (2) b 2 ( 1) x x − . (5) 11 The curve y = f(x) passes through the point (2, −5). Given that f ′(x) = 4x3 − 8x, a find an expression for f(x), (4) b find the coordinates of the points where the curve crosses the x-axis. (4) 12 The curve C with equation y = f(x) is such that d d y x = k − 1 2 x − , x > 0, where k is a constant. Given that C passes through the points (1, −2) and (4, 5), a find the value of k, (5) b show that the normal to C at the point (1, −2) has the equation x + 2y + 3 = 0. (4) C1 INTEGRATION Worksheet C continued PMT