Evaluate the integral \int_0^1 \frac {5-x}{1+x}^2 dx


Evaluate the integral {eq}\int_0^1 \frac {5-x}{1+x}^2 dx {/eq}

The Method of Simple Fractions:

The method of simple fractions to solve an integral is used when the fraction we want to integrate is proper. When the fraction is improper before using the method we must make the division first.

Answer and Explanation:

Transform the integrand

{eq}\displaystyle H=\frac {5-x}{1+x}^2\\ \displaystyle H=\frac {(5-x)^2}{(1+x)^2}\\ \displaystyle H=\frac {x^2-10x+25}{x^2+2x+1}\\ \displaystyle H=\frac {x^2-10x+25}{x^2+2x+1}\\ \displaystyle H=1+ \frac{-12x+24}{x^2+2x+1} {/eq}

Resolve into partial fractions.

{eq}\displaystyle \frac{-12x+24}{x^2+2x+1}= \frac {A}{x+1} +\frac {B}{(x+1)^2}\\ \displaystyle \frac{-12x+24}{x^2+2x+1}= \frac {A(x-1)+B}{ x^2+2x+1}\\ \displaystyle \frac{-12x+24}{x^2+2x+1}= \frac {Ax+A+B}{ x^2(x-1)}\\ \displaystyle A=-12\\ \displaystyle A+B=24 \, \Longrightarrow \, B=36\\ {/eq}

The fraction {eq}\frac{-12x+24}{x^2+2x+1} {/eq} into partial fractions are:

{eq}\displaystyle \Longrightarrow \boxed{ \frac {-12}{x+1} +\frac {36}{(x+1)^2}}\\ {/eq}

Calculate the integral

{eq}\displaystyle I = \int_0^1 \frac {5-x}{1+x}^2 \, dx\\ \displaystyle I = \int_0^1 1+ \frac {-12}{x+1} + \frac {36}{(x+1)^2} \, dx\\ \displaystyle I = \left. x - 12\ln |x+1| - \frac {72}{(x+1)^3} \right|_{0}^{1}\\ \displaystyle I = 1-12\ln |2| - 72 -(-9) \\ \displaystyle I = -12\ln |2| - 62 {/eq}

The result of the integral applying partial fraction method is:

{eq}\displaystyle \Longrightarrow \boxed{ I = -12\ln |2| - 62}\\ {/eq}

Learn more about this topic:

How to Integrate Functions With Partial Fractions

from Math 104: Calculus

Chapter 13 / Lesson 9

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