Question

$\int \frac{2x^2-5x+1}{x^2(x^2-1)}dx=$

• A. $\frac{1}{x}+log\left|\frac{x^5}{(x^2-1)(x+1{)}^2}\right|+$
• B. $\frac{1}{x}+\log \left|\frac{x^5}{(x^2-1)(x+1{)}^3}\right|+c$
• C. $\frac{1}{x}+\log \left|\frac{x^5}{(x^2-1)(x+1{)}^4}\right|+c$
• D. $\frac{1}{x}+\log \left|\frac{x^5}{(x^2-1)(x+1)}\right|+c$

Answer 1

Answer: (B)

$\frac{1}{x}+\log \left|\frac{x^5}{(x^2-1)(x+1{)}^3}\right|+c$

$\int \frac{2x^2-5x+1}{x^2(x^2-1)}dx$

$=\frac{2x^2-5x+1}{x^2(x^2-1)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{(x-1)}+\frac{D}{(x+1)}$

$C=\frac{-2}{2};D=\frac{-8}{+2};B=-1;A=5$

$C=-1;D=-4;B=-1;A=5$

$\int \frac{5}{x}dx-\int \frac{1}{x^2}dx-\int \frac{1}{(x-1)}dx-\int \frac{4}{(x+1)}dx.$

$log\mid x^5\mid +\frac{1}{x}-log\mid (x-1)\mid -log\mid (x+1)\mid dx$

$\frac{1}{x}+log\mid x^{5}1\mid +log\mid (\frac{1}{x}-1)\mid +log\mid \left(\frac{1}{(x+1{)}^4}\right)\mid$

$\frac{1}{x}+log\mid \frac{x^5}{(x^2-1)(x+1{)}^3}\mid +c$