Integralen

Antwoorden bij de opgaven

    1. `F'(x) = 1/(ln(g)) * g^x * ln(g) = g^x`.
    2. Als `f(x) = text(e)^x` dan is `F(x) = 1/ln(text(e)) * text(e)^x + c = text(e)^x + c`.
    3. `int_0^2 h(x) text(d)x = [2x + 0,5text(e)^(2x) * 2]{:(2),(0):} = [2x + text(e)^(2x)]{:(2),(0):} = 3 + text(e)^4`.
    4. De oppervlakte van het gebied begrensd door de grafiek van `h`, de `x`-as en de lijnen `x = 0` en `x = 2`.
    1. `F'(x) = 1/x` als `x > 0` en `F'(x) = 1/x` als `x < 0`.
    2. `G(x) = 4 ln|x + 2| + c`.
    3. `H(x) = 4 ln|2x + 4| * 1/2 + c = 2 ln|2x + 4| + c`.
    1. `F'(x) = 1 * ln(x) + x * 1/x - 1 = ln(x)`.
    2. `F(x) = 1/(ln(g)) * (x ln(x) - x) + c`.
    1. `F(x) = text(e)^(x + 1) + c`
    2. `F(x) = 5x + 3 ln(2) * 2^(4x) * 1/4 = 5x + 3/4 ln(2) * 2^(4x)`
    3. `F(x) = x^2 + ln|x} + c`
    4. `F(x) = 1,5 ln(2x + 1) + c`
    5. `f(x) = (3x ln(3x) - 3x) * 1/3 + c = x ln(3x) - x + c`
    6. `f(x) = 2 text(e)^(-4x) * - 1/4 + c = - 1/(2 text(e)^(-4x)) + c`
    1. `int_ 0^2 3/(3x + 4) text(d)x = [ln|3x + 4|]{:(2),(0):} = ln(10) - ln(4)`.
    2. `int_0^4 0,5^(2x - 1) text(d)x = [1/2 ln(0,5) 0,5^(2x - 1)]{:(4),(0):} = 1/2 ln(0,5) 0,5^7 - 1/2 ln(0,5) 0,5^(-1)`.
    3. `int_1^2 (x^4 + 5x^2)/(x^3) text(d)x = int_1^2 (x + 5/x) text(d)x = [0,5x^2 + 5ln|x|]{:(2),(1):} = 1,5 + 5 ln(2)`.
    4. `int_(0,25)^(text(e)) ln(4x) text(d)x = [x ln(4x) - x]{:(text(e)),(0,25):} = text(e) ln(4text(e)) - text(e) + 0,25`.
    1. `int_0^2 text(e)^x text(d)x = [text(e)^x]{:(2),(0):} = text(e)^2 - 1`.
    2. `int_0^2 pi text(e)^(2x) text(d)x = [1/2 pi text(e)^(2x)]{:(2),(0):} = 1/2 pi (text(e)^(4) - text(e)^2)`.
    3. `pi * 2^2 * 1 + int_(1)^(text(e)^2) pi ln^2(y) text(d)y ~~ 52,71`.
    4. `1 + 2 + text(e)^2 + int_0^2 sqrt(1 + text(e)^(2x)) text(d)x ~~ 17,18`.
    1. `f(x) = 2 1/2` geeft `x = 1/2 vv x = 2`.
      De oppervlakte is `2 1/2 - int_(0,5)^2 (x + 1/x) text(d)x = 2 1/2 - [1/2x^2 + ln(x)]{:(2),(0,5):} = 5/8 - 2ln(2)`.
    2. De inhoud is `int_(0,5)^2 pi * (2,5)^2 text(d)x - int_(0,5)^2 pi (x + 1/x)^2 text(d)x = [6,25pi x]{:(2),(0,5):} - [pi (1/3x^3 + 2ln(x) - 1/x)]{:(2),(0,5):} = 7,25pi - 4 ln(2) - 1,5`.
    3. De omtrek is `1 1/2 + int_(0,5)^2 sqrt(1 + (1 - 1/(x^2))^2) text(d)x ~~ 3,40`.
    1. `[1/2 ln|2x + 1|]{:(1),(0):} = 1/2 ln(3)`.
    2. `[1/(2text(e) + 1) * x^(2text(e) + 1) - 1/2 text(e)^(2x)]{:(1),(0):} = 1/(2text(e) + 1) - 1/2 text(e)^2 + 1/2`.
    3. `[x ln|4x| - x]{:(1),(0,25):} = ln(4) - 0,75`.
    4. `[1/2 x + 2 ln|x|]{:(4),(1):} = 1,5 + 2 ln(4)`.
    5. `[2x + (10)/(ln(10)) * 10^(0,5x)]{:(1),(0):} = 2 + (10)/(ln(10))(sqrt(1) - 1)`.
    6. `[1/(ln(10) * (x ln|3x| - x)]{:(2),(1):} = 1/(ln(10) * (2 ln(6) - 1)`.
    1. `F(x) = 0,5text(e)^(-1/2x) + text(e)x - 0,5`.
    2. `F(x) = x - 1/(ln(10) * (x ln|x| - x) - 1 - 1/(ln(10))`.
    1. `opp(V) = int_2^4 1/2x - 2/x text(d)x = [1/4 x^2 - 2ln|x|]{:(4),(2):} = 3 - 2 ln(2)`.
    2. `l(V) = 2 + 1,5 + int_2^4 sqrt(1 + (1/2 + 1/(x^2))^2) text(d)x ~~ 5,96`.
    3. `I(V) = int_2^4 pi(1/2x - 2/x)^2 text(d)x = pi int_2^4 1/4x^2 - 2 + 4/(x^2) text(d)x = pi[1/12 x^3 - 2x - 4/x]{:(4),(2):} = 1 2/3 pi`.
    1. Doen, gebruik de quotiëntregel.
    2. `opp(V) = int_0^2 g(x) text(d)x = [(2text(e)^x)/(text(e)^x + 1)]{:(2),(0):} = (2text(e)^2)/(text(e)^2 + 1) - 1`.
    3. `[(2text(e)^x)/(text(e)^x + 1)]{:(a),(-a):} = (2text(e)^a)/(text(e)^a + 1) - (2text(e)^(-a))/(text(e)^(-a) + 1)`.
      Herleiden: `(2text(e)^a)/(text(e)^a + 1) - (2text(e)^(-a))/(text(e)^(-a) + 1) * (text(e)^a)/(text(e)^a) = (2text(e)^a - 2)/(text(e)^a + 1) = 1` geeft `text(e)^x = 3` en dus `x = ln(3)`.
    1. Differentieer `F(x)` en laat zien dat `F'(x) = f(x)`.
    2. `opp(V_1) = [-0,5 text(e)^(-x^2)]{:(2),(0):} = 1/2 - 1/(2text(e)^4)`.
    3. `opp(V_1) = [-0,5p text(e)^(-x^2)]{:(2),(0):} = (1/2 - 1/(2text(e)^4)) * p = 10` geeft `p = (20text(e)^4)/(text(e)^4 - 1)`.
    1. `[ln|x| + text(e)^x]{:(2),(1):} = 2ln(2) + text(e)^2 - text(e)`.
    2. `[x + text(e)^x]{:(1),(0):} = text(e)`.
    3. `[2ln|2x + 3|]{:(2),(1):} = 2ln(1,4)`.
    4. `[x - 6 ln|x| - 9/x]{:(2),(1):} = 5,5 - 6ln(2)`.
    5. `[(x - 1)ln(x - 1) - (x - 1)]{:(4),(2):} = 3ln(3) - 2`.
    6. `[1/(3ln(2)) * 2^(3x)]{:(2),(0):} = (2^6 - 1)/(3ln(2))`.
  14. `int_0^6 text(e)^(x/3) - 2x + 5 text(d)x - 1/2 * 5 * 2,5 = [3text(e)^(x/3) - x^2 + 5x]{:(6),(0):} - 6,25 = 3text(e)^2 - 15,25`.
    1. `f'(x) = text(e)^(x) - text(e)^(-x) = 0` geeft `text(e)^(2x) = 1` en dus `x = 0`.
      Min.`f(0) = 2`.
    2. `f(x) = 4` geeft `text(e)^(x) + text(e)^(-x) = 4` en dus `text(e)^(2x) - 4text(e)^(x) + 1 = 0` en `text(e)^x = (4 +- sqrt(12))/2 = 2 +- sqrt(3)`. Oplossing: `ln(2 - sqrt(3)) < x < ln(2 + sqrt(3))`.
    3. `opp(V) = 4 * 2ln(2 + sqrt(3)) - int_(ln(2 - sqrt(3)))^(ln(2 + sqrt(3))) text(e)^(x) + text(e)^(-x) text(d)x = 8ln(2 + sqrt(3)) - [text(e)^(x) - text(e)^(-x)]{:(ln(2 + sqrt(3))),(ln(2 - sqrt(3))):} = 8ln(2 + sqrt(3)) - 4sqrt(3)`.