∫(cotx)dx
To find the integral of cot(x), we can use a technique called integration by substitution
To find the integral of cot(x), we can use a technique called integration by substitution. Here’s a step-by-step solution:
Step 1: Recognize the integral
The integral of cot(x) can be written as ∫cot(x) dx.
Step 2: Rewrite cot(x) using trigonometric identities
We know that cot(x) is the reciprocal of tan(x), so we can rewrite the integral as ∫(1/tan(x)) dx.
Step 3: Simplify the expression
To simplify the expression, we can rewrite tan(x) as sin(x)/cos(x). Now, the integral becomes ∫(1/(sin(x)/cos(x))) dx.
Step 4: Flip the fraction
To make it easier to integrate, we can flip the fraction and change the division to multiplication. Our integral now becomes ∫(cos(x)/sin(x)) dx.
Step 5: Substitute for u
Let’s define u = sin(x). This means that du/dx = cos(x), and dx = du/cos(x).
Step 6: Rewrite the integral using u
Substituting u into our integral, we get ∫(1/u) (du/cos(x)).
Step 7: Simplify the expression
The cos(x) term in the denominator can be canceled out with the du term in the numerator. Our integral simplifies to ∫(1/u) du.
Step 8: Evaluate the integral
The integral of 1/u with respect to u is ln|u|. Therefore, our final solution is ln|u| + C, where C is the constant of integration.
Step 9: Substitute back for u
Since u = sin(x), we substitute back to get ln|sin(x)| + C as the final answer.
So, the integral of cot(x) is ln|sin(x)| + C, where C is the constant of integration.
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