Antiderivative Of Csc(X) Cot(X) Using Substitution Method: Step By Step Guide With Examples

cscx cotx dx

-cscx + C

We can start by using the identity:

csc(x) = 1/sin(x)

cot(x) = cos(x)/sin(x)

So, we have:

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

Now, we can use a substitution to simplify the integral. Let u = sin(x), then du/dx = cos(x) dx. We can rearrange this to get dx = du/cos(x).

Substituting this into the integral, we have:

csc(x) cot(x) dx = (1/u) * (cos(x)/sin(x)) * (du/cos(x))

Simplifying, we get:

csc(x) cot(x) dx = du/u

Now, we can integrate this expression:

∫ csc(x) cot(x) dx

= ∫ du/u

= ln|u| + C

Substituting back in for u, we get:

∫ csc(x) cot(x) dx = ln|sin(x)| + C

Therefore, the antiderivative of csc(x) cot(x) is ln|sin(x)| + C.

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