Question

$\int \frac{dx}{x(x^4-1)}=$

• A. $\frac{1}{4}\log \left|\frac{x^4+1}{x^4}\right|+C$
• B. $\frac{1}{4}\log \left|\frac{x^4}{x^4+1}\right|+C$
• C. $\frac{1}{4}\log |x^4+1|+C$
• D. $\frac{1}{4}\log \left|\frac{x^4}{x^4+2}\right|+C$

Answer 1

The correct option is B $\frac{1}{4}\log \left|\frac{x^4}{x^4+1}\right|+C$

$\int \frac{1}{x(x^4+1)}dx=\frac{1}{4}\int \frac{4x^3}{x^4(x^4+1)}dx$

Put $x^4=t\Rightarrow 4x^{3}dx=dt$

$=\frac{1}{4}\int \frac{1}{t(t+1)}dt=\frac{1}{4}\int (\frac{1}{t}-\frac{1}{t+1})dt$

$=\frac{1}{4}[\log \left|t\right|-\log |t+1|]+c$

$=\frac{1}{4}\log \left|\frac{t}{t+1}\right|+c$

$=\frac{1}{4}\log \left|\frac{x^4}{x^4+1}\right|+c$