Question

If $\int \frac{dx}{x^{22}(x^7-6)}=\frac{1}{K}\{\ln |t|-\frac{t^3}{3}-3t+\frac{3t^2}{2}\}+C$, then which of the following is/are true
(where $K$ is fixed constant and $C$ is constant of integration)

• A. $\frac{K}{1008}=18$
• B. $\frac{K}{1008}=9$
• C. $t=1-\frac{6}{x^7}$
• D. $t=1+\frac{6}{x^7}$

Answer 1

Answer: (B), (C)

$t=1-\frac{6}{x^7}$

Let $I=\int \frac{dx}{x^{22}(x^7-6)}$
$\Rightarrow I=\int \frac{dx}{x^{29}(1-6x^{-7})}$
Put $1-6x^{-7}=t$
$\Rightarrow 42x^{-8}dx=dt$
$\Rightarrow I=\int \frac{1}{x^{21}}\frac{dx}{x^8(1-6x^{-7})}$
$\Rightarrow I=\int \frac{(1-t{)}^3}{6^3}\frac{dt}{42t}$
$\Rightarrow I=\frac{1}{9072}\int \frac{1-3t+3t^2-t^3}{t}dt$
$\Rightarrow I=\frac{1}{9072}\{\ln |t|-3t+\frac{3t^2}{2}-\frac{t^3}{3}\}+C$
$\therefore \frac{K}{1008}=9$ and $t=1-\frac{6}{x^7}$