Index

Integration

Indefinite integral

  1. ?
    ?
    Indefinite integral of a function means finding the anti-derivative.

    xndx=xn+1n+1+C   (for n1)

    x1dx=ln|x|+C

    exdx=ex+C
    Calculate the following integrals:

    (a)   (4x+5)dx

    (b)   (x26x+2)dx

    (c)   (x32x2+5x7)dx

    Solutions:    (a)  =2x2+5x+C;     (b)  =13x33x2+2x+C;     (c)  =14x423x3+52x27x+C
  2. Calculate the following integrals (hint: expand the brackets first).

    (a)   (x+2)(x+4)dx

    (b)   (x1)(x+1)dx

    (c)   (3x+1)2dx

    Solutions:    (a)  =13x3+3x2+8x+C;     (b)  =13x3x+C;     (c)  =3x3+3x2+x+C
  3. Integrate:

    (a)   (6x2+ex)dx

    (b)   (7ex+1x)dx

    (c)   (1x+1x2)dx

    (d)   (x56x3)dx

    Solutions:    (a)  =2x3+ex+C;     (b)  =7ex+ln|x|+C;     (c)  =ln|x|1x+C;     (d)  =16x6+3x2+C
  4. Integrate with respect to x:

    (a)   (x+x+1)dx

    (b)   3xdx

    (c)   (x3+x23)dx

    Solutions:    (a)  =12x2+23x3+x+C;     (b)  =6x+C;     (c)  =34x43+35x53+C
  5. Find the equation of the function which has   f(x)=6x4   and   f(1)=5.
    Solution:    f(x)=3x24x+6
  6. Find the function f knowing that its derivative is f(x)=2x+3 and its graph passes through the point P(2,8).
    Solution:    f(x)=x2+3x2
  7. A function has the value 8 at x=3. Its derivative is f(x)=x2+4x7. Find the equation of this function.
    Solution:    f(x)=13x3+2x27x+2
  8. A curve passes through the point P(1,4) and its gradient function is 9x25. Write down the equation of this curve.
    Solution:    y=3x35x+6
  9. Find y and express it in terms of x (for x>0), given that dydx=1x and y=3 when x=1.
    Solution:    y=lnx+3

Definite integral

  1. ?
    ?
    Definite integral is calculated as:

    abf(x)dx=F(b)F(a)

    Here F is the indefinite integral of f:
    F(x)=f(x)dx
    Evaluate each of the following integrals:

    (a)   12(x2+5) dx

    (b)   02(x3x+2) dx

    (c)   13(x33x2+5) dx

    Solutions:    (a)  =12;     (b)  =6;     (c)  =4
  2. Evaluate each of the following integrals:

    (a)   12x2 dx

    (b)   09x dx

    (c)   01ex dx

    Solutions:    (a)  =12;     (b)  =18;     (c)  =e11.72
  3. Evaluate the integrals:

    (a)   20(x34x) dx

    (b)   02(x34x) dx

    (c)   22(x34x) dx

    Solutions:    (a)  =4;     (b)  =4;     (c)  =0
  4. Find the area of the figure enclosed by the graph of the function f(x)=x2x+1, the x-axis and vertical lines x=1 and x=2.
    Solution:    A=92=4.5
  5. Find the area of the region between the graph of the function f(x)=ex and the x-axis for 0x2. Round the result to three significant figures.
    Solution:    A=6.39
  6. Find the area of the region between the curve y=3x and the x-axis on the interval [1,4].
    Solution:    A=14
  7. Find the area of the region between the curve y=9x2 and the x-axis.
    Solution:    A=36
  8. Find the area of the region enclosed by the graph of the function f(x)=x3+2x2 and the x-axis.
    Solution:    A=43
  9. Find the area of the region enclosed by the curve y=x2 and the straight line y=x+2.
    Solution:    A=92=4.5
  10. Find the area between the curves y=x24x2 and y=4x2.
    Solution:    A=2113
  11. Find the area of the figure between the graphs of the functions f(x)=x21 and g(x)=5x2. Give the result in the exact form.
    Solution:    A=83
  12. Find the area of the region enclosed by the graphs of the functions f(x)=3x and g(x)=x22x.
    Solution:    A=6
  13. Find the area of the region enclosed by the graphs of the functions f(x)=x3 and g(x)=x4.
    Solution:    A=0.55
  14. Find the area of the region enclosed by the curve f(x)=1x and the straight line y=x+52.
    Solution:    A0.489
  15. Find the area of the region enclosed by the curve f(x)=1x2+1 and the straight line y=110.
    Solution:    A1.90
  16. Find the area of the region enclosed by the curves f(x)=4x and g(x)=(x3)2.
    Solution:    A2.55
  17. Consider the functions f(x)=x43x and g(x)=3x2.

    (a)   Write down the intersection points of f and g.

    (b)   Find the area of the region bounded by the graphs of f and g.

    Solutions:    (a)  P1(0.729,2.47), P2(1.51,0.709);     (b)  A6.45
  18. Let f(x)=x3+3x2+x+3.

    (a)   Write down the equation of the tangent to f at x=0.

    (b)   Find the area of the region bounded by the graph of f and this tangent.

    Solutions:    (a)  y=x+3;     (b)  A=6.75
  19. Let f(x)=12x32x2+2x.

    (a)   Write down the equation of the tangent to f at x=0.

    (b)   Find the area of the region bounded by the graph of f and this tangent.

    Solutions:    (a)  y=2x;     (b)  A10.7
  20. Let f(x)=13x4x3.

    (a)   Write down the equation of the normal to f at point A(3,0).

    This normal intersects the graph of the function at A and at another point B.

    (b)   Write down the coordinates of B.

    (c)   Find the area of the region bounded by the graph of f and the normal.

    Solutions:    (a)  y=19x+13;     (b)  B(0.693,0.410);     (c)  Area 4.74
  21. Find the area of the region enclosed by the curve y=x+4, the straight line y=2x and the x-axis.
    Solution:    A7.33
  22. Consider the function f(x)=x24x+6.

    (a)   Write down the equation of the tangent to f at x=0.

    (b)   Write down the equation of the tangent to f at x=3.

    (c)   Write down the intersection point of these tangents.

    (d)   Find the area of the region bounded by the graph of f and both tangents.

    Solutions:    (a)  y=4x+6;     (b)  y=2x3;     (c)  P(1.5,0);     (d)  Area =94=2.25

Trapezoidal formula

  1. ?
    ?
    Trapezoidal formula can be used to approximate the definite integral (or the area):

    A12h(y0+2y1+2y2+2y3++yn)

    Here h=ban
    Let f(x)=14x2+1. Use the trapezoidal formula to find the approximate area of the region between the graph and the horizontal axis on the interval [0,4]. Divide this interval in four parts.
    Solution:    A9.5
  2. Let A=19x dx. Find the approximate value of this integral using the trapezoidal formula.

    (a)   Divide the interval in four parts.

    (b)   Divide the interval in eight parts.

    (c)   Compare both approximations with the exact value.

    Solutions:    (a)  A17.23;     (b)  A17.31;     (c)  A=1713
  3. A table of values of the function y=f(x) is given:
    x 0 0.5 1 1.5 2 2.5 3
    y 1.75 2.25 3.25 2.75 2.5 2 1.25
    Use the trapezoidal formula to find the area of the region under the graph of the function between x=1 and x=3.
    Solution:    A4.75

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