Question

The integral in list 1 can be written as the sum of some functions from List 2. Match the appropriate options from List 2 with those in List 1

Answer 1

A:
$\int \frac{x^2-x+1}{x^3-4x^2+4x}dx=\int [\frac{k_1}{x}+\frac{k_2}{x-2}+\frac{k_3}{(x-2{)}^2}]dx$
Hence, options 1, 2 and 3 are correct for option A.
B. $\int \frac{x^2-1}{x(x-2{)}^3}dx=\int [\frac{k_1}{x}+\frac{k_2}{x-2}+\frac{k_3}{(x-2{)}^2}+\frac{k_4}{(x-2{)}^3}]dx$
Hence, options 1, 2 and 3 are correct for B
C
$\int dx\frac{x^3+1}{x(x-2{)}^2}dx=\int [(\frac{x^3+1}{x(x-2{)}^2}-1)+1]$
$=\int [\left(\frac{x^3+1-x(x-2{)}^2}{x(x-2{)}^2}\right)+1]dx$
$=\int [(\frac{k_1}{x}+\frac{k_2}{x-2}+\frac{k_3}{(x-2{)}^2})+1]dx$
Hence, options 1, 2, 3 and 4 are correct for C
D
$\int \frac{x^5+1}{x(x-2{)}^3}dx=\int [x+k+\frac{g\left(x\right)}{x(x-2{)}^3}]dx,$
$\int [x+k+\frac{k_1}{x}+\frac{k_2}{(x-2)}+\frac{k_3}{(x-2{)}^2}+\frac{k_4}{(x-2{)}^3}]dx$
Hence, options 1, 2, 3 and 4 are correct for D