# Find all the antiderivatives of the following function. Check your work by taking derivatives. q(s)=\csc^{2}s

Find all the antiderivatives of the following function. Check your work by taking derivatives.
$$\displaystyle{q}{\left({s}\right)}={{\csc}^{{{2}}}{s}}$$

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Step 1
Given:
$$\displaystyle{q}{\left({s}\right)}={{\csc}^{{{2}}}{s}}$$
The objective is to find the antiderivatives of the function
Step 2
Considering the given function
$$\displaystyle{q}{\left({s}\right)}={{\csc}^{{{2}}}{s}}$$
Recollecting the derivative formula is
$$\displaystyle{\frac{{{d}}}{{{d}{s}}}}{\left({\cot{{s}}}\right)}=-{{\csc}^{{{2}}}{s}}$$
$$\displaystyle{\frac{{{d}}}{{{d}{s}}}}{\left(-{\cot{{s}}}\right)}={{\csc}^{{{2}}}{s}}$$
On reversing the derivative formula
an antiderivative of $$\displaystyle{q}{\left({s}\right)}={{\csc}^{{{2}}}{s}}\ {i}{s}\ -{\cot{{s}}}$$
$$\displaystyle{q}{\left({s}\right)}=-{\cot{{s}}}+{C}$$
Where,
C is any arbitrary constant
$$\displaystyle{q}{\left({s}\right)}=-{\cot{{s}}}+{C}$$