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int cot(-x)dx

$\int \cot(-x)\,dx$

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Find indefinite integrals using the substitution and by parts

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Q.

\int \cot(-x)\,dx

Rephrasing the Problem: Let's start by rephrasing the ＂Find the indefinite integral of $\cot(-x)$ with respect to $x$ .＂
Applying Odd Function Property: The integral we need to solve is $\int \cot(-x)\,dx$ . We know that $\cot(-x)$ is the same as $-\cot(x)$ because cotangent is an odd function. So, we can rewrite the integral as $-\int \cot(x)\,dx$ .
Integration by Substitution: Now, we need to integrate $-\cot(x)$ . We know that $\cot(x) = \frac{\cos(x)}{\sin(x)}$ , so we can rewrite the integral as $-\int \frac{\cos(x)}{\sin(x)}\,dx$ .
Substitute and Simplify: To integrate $-\cos(x)/\sin(x)$ , we can use a substitution method. Let $u = \sin(x)$ , then $du = \cos(x)dx$ . This substitution simplifies the integral to $-\int \frac{1}{u} du$ .
Final Integration: The integral of $-1/u$ with respect to $u$ is $-\ln|u| + C$ , where $C$ is the constant of integration.
Substitute Back: Now we substitute back $\sin(x)$ for $u$ to get the integral in terms of $x$ . So, the integral becomes $-\ln|\sin(x)| + C$ .
Overall Solution: Therefore, the indefinite integral of $\cot(-x)$ with respect to $x$ is $-\ln|\sin(x)| + C$ .

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