Find all the antiderivatives of the following function. Check your work by taking derivatives. h(z)=-6z^{-7}

remolatg 2021-03-26 Answered
Find all the antiderivatives of the following function. Check your work by taking derivatives.
\(\displaystyle{h}{\left({z}\right)}=-{6}{z}^{{-{7}}}\)

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Expert Answer

Nathalie Redfern
Answered 2021-03-28 Author has 6167 answers
Step 1
To find the antiderivative, we will integrate.
\(\displaystyle\int{h}{\left({z}\right)}{\left.{d}{z}\right.}\)
\(\displaystyle=\int-{6}{z}^{{-{7}}}{\left.{d}{z}\right.}\)
\(\displaystyle=-{6}\int{z}^{{-{7}}}{\left.{d}{z}\right.}\)
\(\displaystyle=-{6}{\frac{{{z}^{{-{7}+{1}}}}}{{-{7}+{1}}}}+{C}\)
\(\displaystyle=-{6}{\frac{{{z}^{{-{6}}}}}{{-{6}}}}+{C}\)
\(\displaystyle{z}^{{-{6}}}+{C}\)
\(\displaystyle={\frac{{{1}}}{{{z}^{{{6}}}}}}+{C}\)
C=integrating constant
Step 2
Then we check it using derivative.
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{z}\right.}}}}{\left({\frac{{{1}}}{{{z}^{{{6}}}}}}+{C}\right)}\)
\(\displaystyle={\frac{{{d}}}{{{\left.{d}{z}\right.}}}}{\left({z}^{{-{6}}}+{C}\right)}\)
\(\displaystyle=-{6}{z}^{{-{6}+{1}}}+{0}\)
\(\displaystyle=-{6}{z}^{{-{7}}}\)
So it shows that the answer is correct.
Answer: \(\displaystyle{\frac{{{1}}}{{{z}^{{{6}}}}}}+{C}\)
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