Demandes associées au chapitre-notebook: "Intégration"

Pour accéder au code relatif à un exemple, double-cliquer sur le triangle du crochet icônique à droite.

Exemple 1

∫(1/((x^2) - 4)) \[DifferentialD]x

Exemple 2

∫((x + 1)/((x^3) + (x^2) - 6 x)) \[DifferentialD]x

Exemple 3

∫((3 x + 5)/((x^3) - (x^2) - x + 1)) 
        \[DifferentialD]x

Exemple 4

∫(((x^4) - (x^3) - x)/((x^3) + (x^2) - 6 x)) 
        \[DifferentialD]x

Exemple 5

∫(1/x) \[DifferentialD]x

Exemple 6

(∫_0^1) x (x - 1) \[DifferentialD]x

Exemple 7

(∫_(-3)^(-2))(1/x) \[DifferentialD]x

Plot[(1/x), {x, -3, -2}, PlotRange -> {-1, 0}, 
      AspectRatio -> Automatic];

(∫_(-1)^2) (1/x) \[DifferentialD]x

Plot[(1/x), {x, -1, 2}, PlotRange -> {-2, 2}, 
      AspectRatio -> Automatic];

(∫_1^2) (1/x) \[DifferentialD]x

(∫_2^1) (1/x) \[DifferentialD]x

Exemple 8

Plot[(1/((x^4) + 1)), {x, 0, π}, AspectRatio -> Automatic];

(∫_0^π) (1/((x^4) + 1)) \[DifferentialD]x

N[%]

Exemple 9

(∫_0^π) (1/(3 + 2 Cos[t])) \[DifferentialD]t

N[%]

Plot[(1/(3 + 2 Cos[t])), {t, 0, π}, 
      AspectRatio -> Automatic, PlotRange -> {0, 1}];

Exemple 10

Show [Graphics [{Text ["\!\(TraditionalForm\`\(TraditionalForm\`\(TraditionalForm\`y = \(b\\ \@\(b\^2 - x\^2\)\)\/a\)\)\)", {-5, 1} {-1.1, 3.2}],
			Text ["\!\(TraditionalForm\`\(TraditionalForm\`y = \(\(-b\)\\ \@\(b\^2 - x\^2\)\)\/a\)\)", {-5, -1} {-1.2, 3}],
			Line[{{5, -5}, {5, 5}}],Circle[{0, 0}, {9, 5}]},Axes -> True]];

(∫_(-a)^a)
        ((∫_(-
                ((b (Sqrt[(a^2) - (x^2)]))/a))^(((b 
                    (Sqrt[(a^2) - (x^2)]))/a)))1 \[DifferentialD]y) 
        \[DifferentialD]x

{Sign [2],  Sign [-2]}

%%  /.  Sign[a]  ->  1

Exemple 11

Needs["Graphics`ImplicitPlot`"]

ImplicitPlot[{(y^2) == x, (y^2) == 4 - x}, {x, -5, 5}, {y, -3, 
        3}, AspectRatio -> Automatic];

Solve[{(y^2) == x, (y^2) == 4 - x}, {x, y}]

(∫_(-(Sqrt[2]))^((Sqrt[2])))(
          (∫_((y^2))^(4 - (y^2)))(1 + 2 x + y) 
              \[DifferentialD]x) \[DifferentialD]y

Exemple 12

(∫_1^2)(Sqrt[(x - 2) (1 - x)]) 
        \[DifferentialD]x

(∫_2^1)(Sqrt[(x - 2) (1 - x)]) 
        \[DifferentialD]x

Plot[(Sqrt[(1 - x) (-2 + x)]), {x, 1, 2}, 
      AspectRatio -> Automatic];

Exemple 13

(∫_3^5)(Sqrt[(x - 3) (4 - x)]) 
        \[DifferentialD]x

Plot[(Sqrt[(x - 3) (4 - x)]), {x, 3, 4}, 
      AspectRatio -> Automatic];

Exemple 14

(∫_(-1)^1) (1/((Sqrt[1 + (x^2)]) + (Sqrt[1 - (x^2)]))) 
        \[DifferentialD]x

N[%]

Plot[(1/((Sqrt[1 + (x^2)]) + (Sqrt[1 - (x^2)]))), {x, -1, 1}, 
      AspectRatio -> Automatic];

(∫_(-1)^1)
        (((Sqrt[ 1 + (x^2)]) - (Sqrt[1 - (x^2)]))/(2 (x^2))) 
        \[DifferentialD]x

N[%]

Exemple 15

(∫_(-1)^1) (1/(Sqrt[(1 - x) (1 + x)])) 
        \[DifferentialD]x

Plot[(1/(Sqrt[(1 - x) (1 + x)])), {x, -1, 1}, 
      AspectRatio -> Automatic];

Exemple 16

(∫_0^∞) Sin[x]/x \[DifferentialD]x

Plot[Sin[x]/x, {x, 0, 50}];

Exemple 17

(∫_0^∞) (1/((x^2) - (a^2))) \[DifferentialD]x

Exemple 18

(∫_0^π) (1/(1 + Cos[a] Cos[x])) 
        \[DifferentialD]x

Exemple 19

NIntegrate[Sin[2 π t]^2, {t, -0.5, 0.5}]

Plot[Sin[2 π t]^2, {t, -0.5, 0.5}, 
      AspectRatio -> Automatic];

Plot[
      Evaluate[Table[
          2 (1 - (r^2)) Sin[2 π t]^2, {r, 0.1, 0.8, 0.1}]], {t, 
        -0.5, 0.5}, AspectRatio -> Automatic];

(NIntegrate[#1, {t, -0.5, 0.5}]&)/@
      Table[2 (1 - (r^2)) Sin[2 π t]^2, {r, 0.1, 0.8, 0.1}]