Question

The value of $\underset{0}{\overset{1}{\int }}\frac{x^a-1}{\ln x}dx,$ (where $a$ is parameter) is equal to

• A. $\ln |a-1|$
• B. $\ln |a+1|$
• C. $a\ln |a+1|$
• D. $a\ln |a-1|$

Answer 1

The correct option is B $\ln |a+1|$

Let $I=\underset{0}{\overset{1}{\int }}\frac{x^a-1}{\ln x}dx\cdots \left(i\right)$
Differentiating w.r.t $a$ keeping $x$ as constant,
$\therefore \frac{dI}{da}=\underset{0}{\overset{1}{\int }}\frac{d}{da}\left(\frac{x^a-1}{\ln x}\right)dx$
$=\underset{0}{\overset{1}{\int }}\frac{x^a\ln x}{\ln x}dx$
$=\underset{0}{\overset{1}{\int }}{x}^{a}dx$
$={\left[\frac{x^{a+1}}{a+1}\right]}_0^1$
$=\frac{1}{a+1}$
Integrating both sides w.r.t.$a,$ we get
$I=\ln |a+1|+c$
From equation $\left(i\right)$,
for $a=0,I=0$
$\Rightarrow 0=\ln 1+c\Rightarrow c=0$
$\therefore I=\ln |a+1|$