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The GenerateConditions->True option of the Integrate command is useful and works properly in most cases. However, there are cases when this command works improperly. Here is one of these: the result of

Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]

is not correct if the integration range includes a zero of Cos[x]-1==01.

The result of

Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

-Cot[b/2] + Cot[\[Pi]/8]

is not better. The same issue in 12.3.1 on Windows 10 with Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True] etc.

Is there a workaround?

The GenerateConditions->True option of the Integrate command is useful and works properly in most cases. However, there are cases when this command works improperly. Here is one of these: the result of

Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]

is not correct if the integration range includes a zero of Cos[x]-1==0.

The result of

Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

-Cot[b/2] + Cot[\[Pi]/8]

is not better. The same issue in 12.3.1 on Windows 10 with Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True] etc.

Is there a workaround?

The GenerateConditions->True option of the Integrate command is useful and works properly in most cases. However, there are cases when this command works improperly. Here is one of these: the result of

Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]

is not correct if the integration range includes a zero of Cos[x]-1.

The result of

Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

-Cot[b/2] + Cot[\[Pi]/8]

is not better. The same issue in 12.3.1 on Windows 10 with Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True] etc.

Is there a workaround?

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user64494
user64494
  • 29.1k
  • 4
  • 29
  • 56

The GenerateConditions->True option of the Integrate command is useful and works properly in most cases. However, there are cases when this command works improperly. Here is one of these cases: the result of

Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]

is not correct if the integration range includes a zero of Cos[x]-1==0.

The result of

Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

-Cot[b/2] + Cot[\[Pi]/8]

is not better. The same issue in 12.3.1 on Windows 10 with Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True] etc.

Is there a workaround?

The GenerateConditions->True option of the Integrate command is useful and works properly in most cases. However, there are cases when this command works improperly. Here is one of these cases: the result of

Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]

is not correct if the integration range includes a zero of Cos[x]-1==0.

The result of

Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

-Cot[b/2] + Cot[\[Pi]/8]

is not better. The same issue in 12.3.1 on Windows 10 with Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True] etc.

Is there a workaround?

The GenerateConditions->True option of the Integrate command is useful and works properly in most cases. However, there are cases when this command works improperly. Here is one of these: the result of

Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]

is not correct if the integration range includes a zero of Cos[x]-1==0.

The result of

Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

-Cot[b/2] + Cot[\[Pi]/8]

is not better. The same issue in 12.3.1 on Windows 10 with Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True] etc.

Is there a workaround?

Source Link
user64494
user64494
  • 29.1k
  • 4
  • 29
  • 56

Incorrect result of integration with GenerateConditions

The GenerateConditions->True option of the Integrate command is useful and works properly in most cases. However, there are cases when this command works improperly. Here is one of these cases: the result of

Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]

is not correct if the integration range includes a zero of Cos[x]-1==0.

The result of

Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

-Cot[b/2] + Cot[\[Pi]/8]

is not better. The same issue in 12.3.1 on Windows 10 with Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True] etc.

Is there a workaround?