∫ cotu du
To solve the integral of cot(u) du, we can use a technique called integration by substitution
To solve the integral of cot(u) du, we can use a technique called integration by substitution. We will perform the following steps:
Step 1: Let’s start by considering the integral ∫ cot(u) du.
Step 2: Notice that the derivative of cot(u) is -cosec^2(u) du. We don’t have exactly that in our integral, but we have a similar term cot(u).
Step 3: To simplify our integral, we can rewrite cot(u) as cos(u)/sin(u). Now our integral becomes ∫ (cos(u)/sin(u)) du.
Step 4: Next, we will substitute a new variable to make the integral easier. Let’s define v = sin(u), so that dv = cos(u) du.
Step 5: Substituting these values, we have ∫ (1/v) dv.
Step 6: Integrating 1/v with respect to v gives us ln|v| + C, where C is the constant of integration.
Step 7: Going back to our original variable, we substitute v with sin(u) to find that ∫ (cot(u) du) = ln|sin(u)| + C.
Therefore, the solution to the integral of cot(u) du is ln|sin(u)| + C, where C represents the constant of integration.
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