∫csc(x)dx
To integrate ∫csc(x)dx, we can use integration by substitution method
To integrate ∫csc(x)dx, we can use integration by substitution method. Let’s go step-by-step:
1. First, identify csc(x) as the reciprocal of sin(x), csc(x) = 1/sin(x).
2. Notice that the derivative of the function inside the denominator is involved in the expression (denominator has sin(x) which is the derivative of -cos(x)). So, we will use u-substitution to simplify the integral.
3. Let u = sin(x) and du = cos(x)dx. Rewrite the integral using u:
∫csc(x)dx = ∫(1/sin(x))dx
Now, substitute for dx:
∫csc(x)dx = ∫(1/u)du
4. Simplify the integral:
∫(1/u)du = ln|u| + C, where C is the constant of integration.
5. Finally, substitute back for u:
∫csc(x)dx = ln|sin(x)| + C
Therefore, the antiderivative or indefinite integral of csc(x) is ln|sin(x)| + C, where C is the constant of integration.
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