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

\(\displaystyle{q}{\left({s}\right)}=-{\cot{{s}}}+{C}\)

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

\(\displaystyle{q}{\left({s}\right)}=-{\cot{{s}}}+{C}\)