Question

Express the function 𝑓(𝑥) = 𝑥 /(𝑥−3)² as partial fractions and hence find ∫ 𝑓(𝑥)𝑑x

Answer 1

Solution

The function 𝑓(𝑥) = 𝑥 /(𝑥−3)² may be represented as sum of partial fractions f(x) = A/(x-3)+B/(x-3)² where A, B are arbitrary constants. From equality 𝑥 /(𝑥−3)² = A/(x-3)+B/(x-3)² we’ll get  

A(x-3)+B = x => A=1, B=3A=3 => 𝑓(𝑥) = 1/(𝑥−3)+3/(𝑥−3)²    

Now ∫ 𝑓(𝑥)𝑑x = ∫ [1/(𝑥−3)+3/(𝑥−3)²]𝑑x = ∫ [1/(𝑥−3)]𝑑x + 3∫ [1/(𝑥−3)²]𝑑x = ln|x-3|-3/(𝑥−3)+C  

Answer

𝑓(𝑥) = 1/(𝑥−3)+3/(𝑥−3)²

∫ 𝑓(𝑥)𝑑x = ln|x-3|-3/(𝑥−3)+C