Integration Guide | How to Find the Integral of csc(x)cot(x) using Substitution

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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