Step-by-Step Guide to Integrating ∫csc(x)dx: Utilizing Integration by Substitution Method

∫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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