Simplifying and Integrating the Function ∫csc(x)cot(x)dx

∫cscxcotxdx

To integrate the function ∫csc(x)cot(x)dx, we can simplify it using trigonometric identities

To integrate the function ∫csc(x)cot(x)dx, we can simplify it using trigonometric identities.

Step 1: Rewrite the integrand
Recall that csc(x) is the reciprocal of sin(x) and cot(x) is the reciprocal of tan(x). Therefore, we can rewrite the integrand as ∫(1/sin(x)) * (cos(x)/sin(x)) dx.

Step 2: Combine the two fractions
To combine the two fractions, multiply the numerators to get cos(x), and multiply the denominators to get sin^2(x). This gives us ∫cos(x)/sin^2(x) dx.

Step 3: Use a substitution
Let’s make the substitution u = sin(x). Taking the derivative of both sides, we get du = cos(x) dx. Rearranging this equation, we have dx = du/cos(x). Plugging this into our integral, we get:

∫(cos(x)/sin^2(x)) dx = ∫(1/u^2) * du

Step 4: Integrate the simplified function
Now that our integral has been simplified, we can continue with the integration. Since the function ∫(1/u^2) du is a standard integral, we can easily integrate it.

∫(1/u^2) du = -1/u + C

Step 5: Substitute back the value of u
We substitute u = sin(x) back into the integrated form, which gives us:

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

Step 6: Simplify the final answer
The final result of the definite integral is given by:

∫csc(x)cot(x)dx = -cot(x) + C

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

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