Using the u-substitution technique to integrate ∫ csc x cot x dx

∫ csc x cot x dx

To integrate the expression ∫ csc x cot x dx, we can use a technique called u-substitution

To integrate the expression ∫ csc x cot x dx, we can use a technique called u-substitution. Let’s go step by step.

Step 1:
First, we need to identify a function and its derivative that can be substituted as u. In this case, let’s set u = cot(x).

To find du (the differential of u), we can differentiate both sides of the equation u = cot(x) with respect to x:
du = -csc^2(x) dx

Step 2:
Now, we need to rearrange the original expression ∫ csc x cot x dx in terms of u. Recall that csc(x) is equal to 1/sin(x), and cot(x) is equal to cos(x)/sin(x).

∫ csc x cot x dx = ∫ (1/sin(x)) * (cos(x)/sin(x)) dx = ∫ (cos(x)/sin^2(x)) dx

Using the identity sin^2(x) = 1 – cos^2(x), we can rewrite the expression as:

∫ (cos(x)/(1 – cos^2(x))) dx

Step 3:
Now, we substitute u and du into the integral:

∫ (cos(x)/(1 – cos^2(x))) dx = -∫ (1/u^2) du = ∫ (-1/u^2) du

Step 4:
Integrating ∫ (-1/u^2) du with respect to u gives us:

∫ (-1/u^2) du = 1/u = 1/cot(x) = tan(x)

So, the final result is:

∫ csc x cot x dx = -tan(x) + C

Where C represents the constant of integration.

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