Mastering Integration: A Step-by-Step Guide to Evaluating ∫csc^2(x) dx

∫(csc²x)dx

To evaluate the integral ∫csc^2(x) dx, we will utilize a technique called integration by substitution

To evaluate the integral ∫csc^2(x) dx, we will utilize a technique called integration by substitution. Here’s how we can proceed:

Step 1: Identify the integral and make a substitution
Let’s rewrite the integral in terms of a trigonometric function that can simplify the expression. We know that csc(x) = 1/sin(x), so by squaring both sides, we can write csc^2(x) as (1/sin(x))^2 or (1/sin^2(x)).

Let’s make a substitution to transform the integral. We can let u = sin(x), so du = cos(x) dx.

Step 2: Rewrite the integral in terms of the new variable
Replace the occurrences of sin(x) and dx in the integral with the new variable u and du:

∫(csc^2(x)) dx = ∫(1/sin^2(x)) dx = ∫(1/u^2) du

Step 3: Evaluate the integral using the new variable
Now we can evaluate the integral with respect to u:

∫(1/u^2) du = ∫u^(-2) du

Step 4: Integrate the expression
Integrate the expression:

∫u^(-2) du = -u^(-1) + C

Step 5: Restore the original variable
Since our substitution was u = sin(x), we need to restore the original variable x:

-u^(-1) + C = -sin^(-1)(x) + C

Therefore, the solution to the integral ∫(csc^2(x)) dx is -cot(x) + C, with C being the constant of integration.

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