Evaluate the integral of \cos^{2} ydy

Lorenzolaji 2021-11-22 Answered
Evaluate the integral of \(\displaystyle{{\cos}^{{{2}}}{y}}{\left.{d}{y}\right.}\)

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Expert Answer

Elizabeth Witte
Answered 2021-11-23 Author has 695 answers
Step 1
Integration is summation of discrete data. The integral is calculated for the functions to find their area, displacement, volume, that occurs due to combination of small data.
Integration is of two types definite integral and indefinite integral. Indefinite integral are defined where upper and lower limits are not given, whereas in definite integral both upper and lower limit are there.
Step 2
The given integrand is \(\displaystyle\int{{\cos}^{{{2}}}{y}}{\left.{d}{y}\right.}\). Here, using trigonometric identities \(\displaystyle{{\cos}^{{{2}}}{y}}\) can be written as \(\displaystyle{\frac{{{1}+{\cos{{2}}}{y}}}{{{2}}}}\).
Substitute the new identity and integrate
\(\displaystyle\int{{\cos}^{{{2}}}{y}}{\left.{d}{y}\right.}=\int{\frac{{{1}+{\cos{{2}}}{y}}}{{{2}}}}{\left.{d}{y}\right.}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}\int{1}+{\cos{{2}}}{y}{\left.{d}{y}\right.}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left[\int{1}{\left.{d}{y}\right.}+\int{\cos{{2}}}{y}{\left.{d}{y}\right.}\right]}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left[{y}+{\frac{{{\sin{{2}}}{y}}}{{{2}}}}\right]}+{C}\)
Therefore, value of \(\displaystyle\int{{\cos}^{{{2}}}{y}}{\left.{d}{y}\right.}\) is equal to \(\displaystyle{\frac{{{1}}}{{{2}}}}{\left[{y}+{\frac{{{\sin{{2}}}{y}}}{{{2}}}}\right]}+{C}\)
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navratitehf
Answered 2021-11-24 Author has 297 answers
Step 1: Use Pythagorean Identities: \(\displaystyle{{\cos}^{{{2}}}{x}}={\frac{{{1}}}{{{2}}}}+{\frac{{{\cos{{2}}}{x}}}{{{2}}}}\).
\(\displaystyle\int{\frac{{{1}}}{{{2}}}}+{\frac{{{\cos{{2}}}{y}}}{{{2}}}}{\left.{d}{y}\right.}\)
Step 2: Use Sum Rule: \(\displaystyle\int{f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}{\left.{d}{x}\right.}=\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}+\int{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle\int{\frac{{{1}}}{{{2}}}}{\left.{d}{y}\right.}+\int{\frac{{{\cos{{2}}}{y}}}{{{2}}}}{\left.{d}{y}\right.}\)
Step 3: Use this rule: \(\displaystyle\int{a}{\left.{d}{x}\right.}={a}{x}+{C}\).
\(\displaystyle{\frac{{{y}}}{{{2}}}}+\int{\frac{{{\cos{{2}}}{y}}}{{{2}}}}{\left.{d}{y}\right.}\)
Step 4: Use Constant Factor Rule: \(\displaystyle\int{c}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle{\frac{{{y}}}{{{2}}}}+{\frac{{{1}}}{{{2}}}}\int{\cos{{2}}}{y}{\left.{d}{y}\right.}\)
Step 5: Use Integration by Substitution on \(\displaystyle\int{\cos{{2}}}{y}{\left.{d}{y}\right.}\).
Let u=2y, du=2dy, then \(\displaystyle{\left.{d}{y}\right.}={\frac{{{1}}}{{{2}}}}{d}{u}\)
Step 6: Using u and du above, rewrite \(\displaystyle\int{\cos{{2}}}{y}{\left.{d}{y}\right.}\).
\(\displaystyle\int{\frac{{{\cos{{u}}}}}{{{2}}}}{d}{u}\)
Step 7: Use Constant Factor Rule: \(\displaystyle\int{c}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle{\frac{{{1}}}{{{2}}}}\int{\cos{{u}}}{d}{u}\)
Step 8: Use Trigonometric Integration: the integral of \(\displaystyle{\cos{{u}}}\ {i}{s}\ {\sin{{u}}}\).
\(\displaystyle{\frac{{{\sin{{u}}}}}{{{2}}}}\)
Step 9: Substitute u=2y back into the original integral.
\(\displaystyle{\frac{{{\sin{{2}}}{y}}}{{{2}}}}\)
Step 10: Rewrite the integral with the completed substitution.
\(\displaystyle{\frac{{{y}}}{{{2}}}}+{\frac{{{\sin{{2}}}{y}}}{{{4}}}}\)
Step 11: Add constant.
\(\displaystyle{\frac{{{y}}}{{{2}}}}+{\frac{{{\sin{{2}}}{y}}}{{{4}}}}+{C}\)
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