Question

If $\int \frac{1-x^7}{x(1+x^7)}dx=A\ln |x|+B\ln |1+x^7|+C$, then which of the following is/are true (where $A,B$ are fixed constants and $C$ is constant of integration)

• A. $A=1$
• B. $A=-1$
• C. $B=-\frac{2}{7}$
• D. $B=\frac{2}{7}$

Answer 1

Answer: (A), (C)

$B=-\frac{2}{7}$

$I=\int \frac{1-x^7}{x(1+x^7)}dx$
$\Rightarrow I=\int \frac{2-(1+x^7)}{x(1+x^7)}dx$
$\Rightarrow I=\int \frac{2dx}{x^8(\frac{1}{x^7}+1)}-\int \frac{dx}{x}$
Put $\frac{1}{x^7}+1=t$
$\Rightarrow -7x^{-8}dx=dt$
$\therefore I=\int \frac{2dt}{-7t}-\ln |x|+C$
$\Rightarrow I=-\frac{2}{7}\ln |1+\frac{1}{x^7}|-\ln |x|+C$
$\Rightarrow I=-\frac{2}{7}\ln |x^7+1|+2\ln |x|-\ln |x|+C$
$\Rightarrow I=\ln |x|-\frac{2}{7}\ln |x^7+1|$
$\therefore A=1$ and $B=-\frac{2}{7}$