To find the antiderivatives of the function [tex]\( f(x) = -2 \csc^2(x) \)[/tex], we follow these steps:
1. Understanding the Function:
The function we need to find the antiderivative for is [tex]\( f(x) = -2 \csc^2(x) \)[/tex].
2. Recalling the Antiderivative of [tex]\(\csc^2(x)\)[/tex]:
The antiderivative of [tex]\(\csc^2(x)\)[/tex] is a known result from calculus:
[tex]\[
\int \csc^2(x) \, dx = -\cot(x) + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
3. Applying the Constant Multiple Rule:
When finding the antiderivative of a function multiplied by a constant, we can factor out the constant and find the antiderivative of the remaining function:
[tex]\[
\int -2 \csc^2(x) \, dx = -2 \int \csc^2(x) \, dx
\][/tex]
4. Substituting the Known Result:
Substitute the antiderivative of [tex]\(\csc^2(x)\)[/tex] from step 2 into the integral:
[tex]\[
-2 \int \csc^2(x) \, dx = -2 \left( -\cot(x) + C \right)
\][/tex]
5. Simplifying the Expression:
Distribute the [tex]\( -2 \)[/tex]:
[tex]\[
-2 \left( -\cot(x) + C \right) = 2 \cot(x) - 2C
\][/tex]
Since [tex]\( C \)[/tex] is an arbitrary constant, multiplying it by [tex]\( -2 \)[/tex] yields another constant which we'll denote as [tex]\( C' \)[/tex]:
[tex]\[
2 \cot(x) + C'
\][/tex]
6. Final Answer:
The general antiderivative of [tex]\( f(x) = -2 \csc^2(x) \)[/tex] is:
[tex]\[
F(x) = 2 \cot(x) + C
\][/tex]
where [tex]\( C \)[/tex] is an arbitrary constant of integration.
Hence, the antiderivatives of [tex]\( f(x) = -2 \csc^2(x) \)[/tex] are:
[tex]\[
F(x) = 2 \cot(x) + C
\][/tex]
