Simplifying the Expression and Integrating csc(x) cot(x) dx using Substitution

cscx cotx dx

To integrate csc(x) cot(x) dx, we can use the technique of substitution

To integrate csc(x) cot(x) dx, we can use the technique of substitution.

Let’s start by simplifying the expression. Remember that csc(x) is the reciprocal of sin(x) and cot(x) is the reciprocal of tan(x).

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

Now, let’s make a substitution. Let u = sin(x), so du = cos(x) dx.

We can rewrite the integral in terms of u:

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

Now, let’s integrate with respect to u:

∫ (1/u^2) du = -u^(-1)

Substituting back for u:

– u^(-1) = – sin^(-1)(x)

Therefore, the integral of csc(x) cot(x) dx is – sin^(-1)(x) + C, where C is the constant of integration.

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