# Evaluate the integral. \int_2^3 \frac{7x + 15}{x^2 + 5x + 0} dx

## Question:

Evaluate the integral.

{eq}\int_2^3 \frac{7x + 15}{x^2 + 5x + 0} dx {/eq}

## Simple fractions:

The method of decomposition into simple fractions consists in breaking down a ratio of polynomials into a sum of fractions of polynomials of lesser degree. It is mainly used in the calculation of integrals as the given exercise.

Apply partial fraction method.

{eq}\displaystyle \frac{7x+15}{x^2+5x}\\ \displaystyle \frac{7x+15}{x(x+5)}= \frac {A}{x} +\frac {B}{x+5}\\ \displaystyle \frac{7x+15}{x(x+5)}= \frac {A(x+5)+Bx)}{x(x+5)} \\ \displaystyle \frac{7x+15}{x(x+5)}= \frac {Ax+5A+Bx}{ x(x+5)} \\ \displaystyle \frac{7x+15}{x(x+5)}= \frac {x(A+B)+5A}{ x(x+5)} \\ \displaystyle 5A=15 \,\, \Longrightarrow \,\, A=3\\ \displaystyle A+B=7 \,\, \Longrightarrow \,\, B=4\\ \displaystyle \frac{7x+15}{x(x+5)}= \frac {3}{x} +\frac {4}{x+5} {/eq}

Calculate the integral

{eq}\displaystyle I = \int_2^3 \frac{7x + 15}{x^2 + 5x + 0} \, dx \\ \displaystyle I = \int_2^3 \frac {3}{x} +\frac {4}{x+5} \, dx \\ \displaystyle I = \left. 3\ln |x| +4\ln |x+5| \right|_{2}^{3}\\ \displaystyle I = 3\ln 3 +4\ln 8 -(3\ln 2 +4\ln 7) \\ \displaystyle I = 3\ln 3 +4\ln 8 -3\ln 2 -4\ln 7 \\ \displaystyle I = \ln 3^3 +\ln 8^4 -\ln 2^3 -\ln 7^4 \\ \displaystyle I = \ln \frac{3^3 8^4}{2^3 7^4} \\ \displaystyle I = \ln \frac{3^3 8^3}{7^4} \\ \displaystyle I = \ln \frac{13824}{2401} \\ \displaystyle I \approx 1.75 {/eq}

The result of the integral applying partial fraction method is:

{eq}\displaystyle \Longrightarrow \boxed{1.75}\\ {/eq} 