express the integrand as a sum of partial fractions and evaluate the integral.\n int_{5}^{9}\frac{y}{y^{2}-y…

express the integrand as a sum of partial fractions and evaluate the integral.\n int_{5}^{9}\frac{y}{y^{2}-y - 2}dy \nexpress the integrand as a sum of partial fractions.\n \frac{y}{y^{2}-y - 2}=square \n(simplify your answer. use integers or fractions for any numbers in the expression.)\nevaluate the integral.\n int_{5}^{9}\frac{y}{y^{2}-y - 2}dy=square \n(use parentheses to clearly denote the argument of each function.)

express the integrand as a sum of partial fractions and evaluate the integral.\n int_{5}^{9}\frac{y}{y^{2}-y - 2}dy \nexpress the integrand as a sum of partial fractions.\n \frac{y}{y^{2}-y - 2}=square \n(simplify your answer. use integers or fractions for any numbers in the expression.)\nevaluate the integral.\n int_{5}^{9}\frac{y}{y^{2}-y - 2}dy=square \n(use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Factor the denominator

Factor $y^{2}-y - 2=(y - 2)(y+1)$.

Step2: Set up partial - fraction decomposition

Let $\frac{y}{y^{2}-y - 2}=\frac{A}{y - 2}+\frac{B}{y + 1}$. Then $y=A(y + 1)+B(y - 2)$.

Step3: Solve for A and B

If $y = 2$, then $2=A(2 + 1)+B(2 - 2)$, so $A=\frac{2}{3}$. If $y=-1$, then $-1=A(-1 + 1)+B(-1 - 2)$, so $B=\frac{1}{3}$. So $\frac{y}{y^{2}-y - 2}=\frac{2/3}{y - 2}+\frac{1/3}{y + 1}$.

Step4: Evaluate the integral

$\int_{5}^{9}\frac{y}{y^{2}-y - 2}dy=\int_{5}^{9}(\frac{2/3}{y - 2}+\frac{1/3}{y + 1})dy$ $=\frac{2}{3}\int_{5}^{9}\frac{1}{y - 2}dy+\frac{1}{3}\int_{5}^{9}\frac{1}{y + 1}dy$ $=\frac{2}{3}[\ln|y - 2|]{5}^{9}+\frac{1}{3}[\ln|y + 1|]{5}^{9}$ $=\frac{2}{3}(\ln(9 - 2)-\ln(5 - 2))+\frac{1}{3}(\ln(9 + 1)-\ln(5 + 1))$ $=\frac{2}{3}(\ln7-\ln3)+\frac{1}{3}(\ln10-\ln6)$ $=\frac{2}{3}\ln\frac{7}{3}+\frac{1}{3}\ln\frac{10}{6}$ $=\frac{2}{3}\ln\frac{7}{3}+\frac{1}{3}\ln\frac{5}{3}$ $=\ln((\frac{7}{3})^{\frac{2}{3}})+\ln((\frac{5}{3})^{\frac{1}{3}})$ $=\ln((\frac{7}{3})^{\frac{2}{3}}(\frac{5}{3})^{\frac{1}{3}})$ $=\ln(\frac{7^{\frac{2}{3}}\times5^{\frac{1}{3}}}{3})$

Answer:

$\frac{y}{y^{2}-y - 2}=\frac{2/3}{y - 2}+\frac{1/3}{y + 1}$; $\int_{5}^{9}\frac{y}{y^{2}-y - 2}dy=\ln(\frac{7^{\frac{2}{3}}\times5^{\frac{1}{3}}}{3})$