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

Evaluate the integrals $$\displaystyle\int{\frac{{{4}}}{{{y}^{{{2}}}}}}{\left.{d}{y}\right.}$$

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Eprint

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
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: $$\displaystyle\int{x}^{{{n}}}{\left.{d}{x}\right.}={\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}}+{C}$$.
$$\displaystyle-{\frac{{{4}}}{{{y}}}}$$
$$\displaystyle-{\frac{{{4}}}{{{y}}}}+{C}$$