∫ cscxcotx dx
To evaluate the integral ∫ csc(x) cot(x) dx, we can use a trigonometric identity to simplify the expression
To evaluate the integral ∫ csc(x) cot(x) dx, we can use a trigonometric identity to simplify the expression.
The identity we’ll use is:
csc(x) = 1/sin(x)
cot(x) = cos(x)/sin(x)
Substituting these values into the integral, we have:
∫ (1/sin(x)) * (cos(x)/sin(x)) dx
Simplifying the expression, we get:
∫ (cos(x)/(sin^2(x))) dx
Now, we can use a substitution method to simplify the integral further. Let’s substitute u = sin(x), so that du = cos(x) dx. Rearranging this equation, we have dx = du / cos(x).
Substituting these values into the integral, we get:
∫ (1/u^2) (du/cos(x))
Simplifying, we have:
∫ du/u^2
This integral can be evaluated as:
-1/u + C
Substituting back u = sin(x), we get:
-1/sin(x) + C
Therefore, the integral of csc(x) cot(x) dx is:
-1/sin(x) + C, where C represents the constant of integration.
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