EXAMPLE 6 Evaluate the following integral.cot(x) dxSOLUTION First we write cotangent in terms of sine and cosine:cot(x) dx = cos(x)sin(x) dx.This suggests that we should substitute u = sin(x), since then du = dx and so:cot(x) dx = cos(x)sin(x) dx = duu = ln(|u|) + C = + C (in terms of x).
Question
EXAMPLE 6 Evaluate the following integral.cot(x) dxSOLUTION First we write cotangent in terms of sine and cosine:cot(x) dx = cos(x)sin(x) dx.This suggests that we should substitute u = sin(x), since then du = dx and so:cot(x) dx = cos(x)sin(x) dx = duu = ln(|u|) + C = + C (in terms of x).
Solution
The text you provided is a step-by-step solution to evaluate the integral of cot(x) dx. Here's the breakdown:
- Rewrite cotangent in terms of sine and cosine: cot(x) dx = cos(x)/sin(x) dx.
- Substitute u = sin(x). This is suggested because the derivative of sin(x) is cos(x), which is present in the integral.
- With the substitution, the integral becomes du/u.
- The integral of du/u is ln|u| + C, where C is the constant of integration.
- Substitute back u = sin(x) to get the final answer in terms of x: ln|sin(x)| + C.
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