Points: 0 of 1 Save Find all antiderivatives for the following function. Check your work by taking the derivative. \[ f(x)=7 \csc ^{2} x \] The antiderivatives of $f(x)=7 \csc ^{2} x$ are $F(x)=\square$.

Step 1 ：We are given the function \(f(x)=7 \csc ^{2} x\). We need to find all antiderivatives of this function.

Step 2 ：The antiderivative of a function is the integral of that function. So, we need to find the integral of \(f(x)=7 \csc ^{2} x\).

Step 3 ：The integral of \(\csc ^{2} x\) is \(-\cot x\). Therefore, the integral of \(7 \csc ^{2} x\) is \(-7\cot x\).

Step 4 ：However, the integral also includes a constant of integration, which we'll denote as \(C\). So the most general antiderivative of \(f(x)\) is \(F(x) = -7\cot x + C\).

Step 5 ：We can check this by taking the derivative of \(F(x)\) and seeing if we get \(f(x)\) back.

Step 6 ：Taking the derivative of \(F(x) = -7\cot x + C\), we get \(f(x) = 7 \csc ^{2} x\). This confirms that \(F(x)\) is an antiderivative of \(f(x)\).

Step 7 ：\(\boxed{F(x) = -7\cot x + C}\) is the final answer, where \(C\) is an arbitrary constant.