## integral of cscsxcotx

### To find the integral of csc(x)cot(x), we can use integration by substitution

To find the integral of csc(x)cot(x), we can use integration by substitution. Let’s go through the steps:

Step 1: Identify the parts of the expression that can be simplified. In this case, we can use the trigonometric identity csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x).

Step 2: Rewrite the expression using the identified trigonometric identities. We have:

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

Step 3: Simplify the expression further:

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

Step 4: Now, we can proceed with the integration. Let’s denote sin(x) as u, then differentiate both sides with respect to x to find du:

du/dx = cos(x), which gives us du = cos(x) dx.

Step 5: Substitute the variables in the integral expression:

∫(cos(x))/(sin(x))^2 dx = -∫1/u^2 du.

Step 6: Integrate the expression with respect to u:

-∫1/u^2 du = -(-1/u) + C = 1/u + C,

where C is the constant of integration.

Step 7: Substitute back u with sin(x):

1/u + C = 1/sin(x) + C.

So, the final result of the definite integral of csc(x)cot(x) is 1/sin(x) + C, where C is the constant of integration.

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