Evaluate the following. intfrac{(10y+11)dy}{4y^2-4y+5}

Question
Trigonometry
asked 2021-02-25
Evaluate the following.
\(\displaystyle\int{\frac{{{\left({10}{y}+{11}\right)}{\left.{d}{y}\right.}}}{{{4}{y}^{{2}}-{4}{y}+{5}}}}\)

Answers (1)

2021-02-26
The given integral is, \(\displaystyle\int{\frac{{{\left({10}{y}+{11}\right)}{\left.{d}{y}\right.}}}{{{4}{y}^{{2}}-{4}{y}+{5}}}}\)
First, complete the square in the denominator as follows:
\(\displaystyle\int{\frac{{{\left({10}{y}+{11}\right)}}}{{{4}{y}^{{2}}-{4}{y}+{5}}}}{\left.{d}{y}\right.}=\int{\frac{{{\left({10}{y}+{11}\right)}}}{{{4}{\left({y}-{\frac{{{1}}}{{{2}}}}\right)}^{{2}}+{4}}}}{\left.{d}{y}\right.}\)
Apply u-substitution, \(\displaystyle{u}={y}-{\frac{{{1}}}{{{2}}}}\)
\(\displaystyle\int{\frac{{{\left({10}{y}+{11}\right)}}}{{{4}{\left({y}-{\frac{{{1}}}{{{2}}}}\right)}^{{2}}+{4}}}}{\left.{d}{y}\right.}=\int{\frac{{{5}{u}+{8}}}{{{2}{\left({u}^{{2}}+{1}\right)}}}}{d}{u}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}\int{\frac{{{5}{u}+{8}}}{{{\left({u}^{{2}}+{1}\right)}}}}{d}{u}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left(\int{\frac{{{5}{u}}}{{{\left({u}^{{2}}+{1}\right)}}}}{d}{u}+\int{\frac{{{8}}}{{{\left({u}^{{2}}+{1}\right)}}}}{d}{u}\right)}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left({\frac{{{5}}}{{{2}}}}{\ln}{\left|{u}^{{2}}+{1}\right|}+{8}{{\tan}^{{-{1}}}{\left({u}\right)}}\right)}\)
Substitute back, \(\displaystyle{u}={y}-{\frac{{{1}}}{{{2}}}}\)
\(\displaystyle{\frac{{{1}}}{{{2}}}}{\left({\frac{{{5}}}{{{2}}}}{\ln}{\left|{u}^{{2}}+{1}\right|}+{8}{{\tan}^{{-{1}}}{\left({u}\right)}}\right)}={\frac{{{1}}}{{{2}}}}{\left({\frac{{{5}}}{{{2}}}}{\ln}{\left|{\left({y}-{\frac{{{1}}}{{{2}}}}\right)}^{{2}}+{1}\right|}+{8}{{\tan}^{{-{1}}}{\left({y}-{\frac{{{1}}}{{{2}}}}\right)}}\right)}\)
\(\displaystyle={\frac{{{1}}}{{{4}}}}{\left({5}{\ln}{\left|{y}^{{2}}-{y}+{\frac{{{5}}}{{{4}}}}\right|}+{16}{{\tan}^{{-{1}}}{\left({y}-{\frac{{{1}}}{{{2}}}}\right)}}\right)}+{C}\)
0

Relevant Questions

asked 2020-10-25
Evaluate the following.
\(\displaystyle\int{\frac{{{{\cos}^{{5}}{\left({3}{z}\right)}}{\left.{d}{z}\right.}}}{{{{\sin}^{{2}}{\left({3}{z}\right)}}}}}\)
asked 2021-02-09
Evaluate the following.
\(\displaystyle\int{\frac{{{\sin{\theta}}{\left({\cos{\theta}}+{4}\right)}{d}\theta}}{{{1}+{{\cos}^{{2}}\theta}}}}\)
asked 2021-02-24
Evaluate the following.
\(\displaystyle\int{{\sin}^{{2}}{x}}\cdot{\tan{{x}}}{\left.{d}{x}\right.}\)
asked 2021-03-09
Evaluate the following
\(\displaystyle\int{{\csc}^{{6}}{u}}{d}{u}\)
asked 2020-12-15
Evaluate the following.
\(\displaystyle\int{\cos{\beta}}{\left({1}-{\cos{{2}}}\beta\right)}^{{3}}{d}\beta\)
asked 2021-02-08
Explain how we can evaluate the expression \(\displaystyle{\cos{{\left({\arctan{{\left({\frac{{{v}}}{{{a}}}}\right)}}}\right)}}}\) and what the evaluation is in terms of v and a.
asked 2021-02-09
The angle \(\displaystyle\theta\) is in the fourth quadrant and \(\displaystyle{\cos{\theta}}={\frac{{{2}}}{{{7}}}}\). Find the exact value of the remaining five trigonometric functions.
asked 2020-12-25
Find the value of each trigonometric ratio to the nearest ten-thousandth
\(\displaystyle{{\tan{{50}}}^{\circ}}\)
asked 2020-10-18
Determine the exact value of expression.
\(\displaystyle{\sec{{\left({210}\right)}}}\times{\cot{{\left({300}\right)}}}+{\sin{{\left({225}\right)}}}\)
asked 2020-12-25
Determine the exact value of expression.
\(\displaystyle{\tan{{\left({60}\right)}}}\times{3}{\sin{{\left({90}\right)}}}-{\sin{{\left({315}\right)}}}\)
...