Question

If $\int \frac{1}{x[{\left(\log x\right)}^2+4\log x-1]}$ integrated gives $=\frac{1}{2\sqrt{\left(5\right)}}\log \frac{(t+2)-\sqrt{5}}{(t+2)+\sqrt{\left(5\right)}}$ , then find t.

• A. $\log x$
• B. $\ln x$
• C. $x$
• D. $\frac{1}{\log x}$

Answer 1

Answer: (A)

$\log x$

Substitute $\log x=t\therefore \frac{1}{x}dx=dt$ $\therefore I=\int \frac{1}{t^2+4t-1}dt=\int \frac{dt}{{(t+2)}^2-{\left(\sqrt{5}\right)}^2}$ $=\frac{1}{2\sqrt{\left(5\right)}}\log \frac{(t+2)-\sqrt{5}}{(t+2)+\sqrt{\left(5\right)}}$ where $t=\log x.$