Integral of csc(x)cot(x) | u-substitution method explained

Integral of cscxcotx

To find the integral of csc(x)cot(x), we can use a method called u-substitution

To find the integral of csc(x)cot(x), we can use a method called u-substitution.

Let’s start by analyzing the given expression:

∫csc(x)cot(x)dx

First, we can rewrite csc(x)cot(x) as cos(x)/sin^2(x). Thus, we can rewrite the integral as:

∫(cos(x)/sin^2(x))dx

Now, let’s apply u-substitution. We’ll substitute u = sin(x), which means that du = cos(x)dx. We can rearrange the integral to have du and u, making it easier to solve:

∫(1/u^2)du

Now, let’s integrate the expression:

∫(1/u^2)du = -1/u + C

However, we need to convert the expression back in terms of x, rather than u. Recall that u = sin(x). Thus, we substitute u back in:

-1/u + C = -1/sin(x) + C = -cosec(x) + C

Therefore, the integral of csc(x)cot(x) is -cosec(x) + C, where C is the constant of integration.

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