Solving the Integral of csc(x)dx using Integration by Substitution: Step-by-Step Guide

∫cscxdx

To solve the integral of csc(x)dx, we can use a technique called integration by substitution

To solve the integral of csc(x)dx, we can use a technique called integration by substitution.

Step 1: Identify a substitution

Let’s make the substitution u = sin(x). This means that du = cos(x)dx.

Step 2: Substitute

Using the substitution u = sin(x), we can rewrite the integral as:

∫ csc(x)dx = ∫ (1/sin(x))dx = ∫ (1/u)du

Step 3: Evaluate the new integral

The integral of (1/u)du is a simpler integral to solve. Integrating, we get:

∫ (1/u)du = ln|u| + C

Step 4: Substitute back

Now, we substitute back u = sin(x) into our result:

ln|sin(x)| + C

So, the integral of csc(x)dx is ln|sin(x)| + C, where C is the constant of integration.

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