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

Dillard 2021-04-26 Answered
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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Expert Answer

odgovoreh
Answered 2021-04-28 Author has 24146 answers
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
\(\displaystyle{q}{\left({s}\right)}=-{\cot{{s}}}+{C}\)
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