Why is the absolute value of sinx used in the integral of cotx?

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Thread starter Vividly
Start date Jan 13, 2022
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Integral

In summary, the integral of cotx is Ln|sinx| + c. To solve it, you can use substitution or integration by parts. It can be simplified to Ln|sinx| + c. It is valid for all values of x except for x = nπ. The integral is the inverse of the derivative of cotx, so its graph is the antiderivative of the graph of cotx.

Jan 13, 2022

Vividly

TL;DR Summary: Why is there a absolute value sign on sinx?

Why is there a absolute value sign on sinx? Does it have to do with the domain of cot x and sin x?

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Jan 13, 2022

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It has to do with the domain of the logarithm. ##\ln \sin x ## isn't defined whenever ##\sin x \leq 0.##

1. What is the integral of cotx?

The integral of cotx is equal to Ln|sinx|+c, where Ln|sinx| represents the natural logarithm of the absolute value of sinx and c is the constant of integration.

2. How do you derive the integral of cotx?

To derive the integral of cotx, we use the substitution method. Let u = sinx, then du = cosx dx. The integral then becomes ∫cotx dx = ∫cotx (cosx/cosx) dx = ∫(1/u) du = Ln|u|+c = Ln|sinx|+c.

3. Can the integral of cotx be simplified further?

No, the integral of cotx cannot be simplified further. It is in its most simplified form as Ln|sinx|+c.

4. What is the domain and range of the integral of cotx?

The domain of the integral of cotx is all real numbers except for multiples of π/2, since cotx is undefined at those points. The range is also all real numbers, as the natural logarithm of any positive number is a real number.

5. Can the integral of cotx be used to solve real-world problems?

Yes, the integral of cotx can be used to solve real-world problems in physics, engineering, and other fields that involve periodic functions. It can also be used to find the area under the curve of a cotangent function.