Evaluate the integrals \int \frac{4}{y^{2}}dy

druczekq4 2021-11-22 Answered
Evaluate the integrals \(\displaystyle\int{\frac{{{4}}}{{{y}^{{{2}}}}}}{\left.{d}{y}\right.}\)

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Expert Answer

Eprint
Answered 2021-11-23 Author has 489 answers

Step 1
we have to evaluate the integral \(\displaystyle\int{\frac{{{4}}}{{{y}^{{{2}}}}}}{\left.{d}{y}\right.}\)
let the given integral be I.
therefore,
\(\displaystyle{I}=\int{\frac{{{4}}}{{{y}^{{{2}}}}}}{\left.{d}{y}\right.}\)
\(\displaystyle=\int{4}{y}^{{-{2}}}{\left.{d}{y}\right.}\)
\(\displaystyle={4}\int{y}^{{-{2}}}{\left.{d}{y}\right.}\)
Step 2
as we know that \(\displaystyle\int{y}^{{{n}}}{\left.{d}{y}\right.}={\frac{{{y}^{{{n}+{1}}}}}{{{n}+{1}}}}\)
therefore,
\(\displaystyle{I}={4}\int{y}^{{-{2}}}{\left.{d}{y}\right.}\)
\(\displaystyle={4}{\left({\frac{{{y}^{{-{2}+{1}}}}}{{-{2}+{1}}}}\right)}+{C}\)
\(\displaystyle={4}{\left({\frac{{{y}^{{-{1}}}}}{{-{1}}}}\right)}+{C}\)
\(\displaystyle=-{4}{y}^{{-{1}}}+{C}\)
\(\displaystyle=-{\frac{{{4}}}{{{y}}}}+{C}\)
where C is constant of integration.
Step 3
therefore the value of the given integral \(\displaystyle\int{\frac{{{4}}}{{{y}^{{{2}}}}}}{\left.{d}{y}\ \right.}{i}{s}{\frac{{-{4}}}{{{y}}}}+{C}\)

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Drood1980
Answered 2021-11-24 Author has 1904 answers
Step 1: Use Constant Factor Rule: \(\displaystyle\int{c}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle{4}\int{\frac{{{1}}}{{{y}^{{{2}}}}}}{\left.{d}{y}\right.}\)
Step 2: Use Power Rule: \(\displaystyle\int{x}^{{{n}}}{\left.{d}{x}\right.}={\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}}+{C}\).
\(\displaystyle-{\frac{{{4}}}{{{y}}}}\)
Step 3: Add constant.
\(\displaystyle-{\frac{{{4}}}{{{y}}}}+{C}\)
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