Question

i) Express the following as the sum of partial fractions. 4/(1+3x)(1+x)² ii) Hence, integrate the function between 0 and 2. Leave your answer in exact form.

Answer 1

To express the fraction as the sum of partial fractions, we need to use the method of partial fraction decomposition and factor the denominator. We can then decompose the fraction into three fractions with different denominators. Finally, we can integrate the function between the given limits.

Step-by-step explanation:

To express the given fraction as the sum of partial fractions, we need to use the method of partial fraction decomposition. The given fraction is 4/[(1+3x)(1+x)²]. We start by factoring the denominator as (1+3x)(1+x)².

Since we have a quadratic factor (1+x)², we can decompose the fraction as:

A/(1+3x) + B/(1+x) + C/(1+x)².

To determine the values of A, B, and C, we can multiply both sides of the equation by the denominator and simplify the resulting equation. After finding the values of A, B, and C, we can integrate the function between 0 and 2.