SOLUTION 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 =
Question
SOLUTION 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 =
Solution 1
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Solution 2
cos(x) dx.
Step 1: Let u = sin(x).
Step 2: Then, du = cos(x) dx.
Step 3: Substitute u and du into the integral, replacing sin(x) and cos(x) dx, respectively.
Step 4: The integral becomes ∫ du/u, which is a standard form.
Step 5: The antiderivative of 1/u is ln|u|, so the integral becomes ln|sin(x)| + C, where C is the constant of integration.
So, the integral of cot(x) dx is ln|sin(x)| + C.
Similar Questions
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).
Verify the identity.cos(u) sec(u)tan(u) = cot(u)Use a Reciprocal Identity to rewrite the expression, and then simplify.
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