Question

$\int \frac{dx}{x(x+1)(\ln (x+1)-\ln x{)}^{11}}$ equals
where $C$ is constant of integration.

• A. $\frac{1}{10(\ln (x+1)-\ln x{)}^{10}}+C$
• B. $\frac{(\ln (x+1)-\ln x{)}^{10}}{10}+C$
• C. $\frac{1}{11(\ln (x+1)-\ln x{)}^{11}}+C$
• D. $\frac{(\ln (x+1)-\ln x{)}^{11}}{11}+C$

Answer 1

Answer: (A)

Let $I=\int \frac{dx}{x(1+x)(\ln (x+1)-\ln x{)}^{11}}$
Put $\ln (x+1)-\ln x=t\Rightarrow \frac{dx}{x(1+x)}=-dt$
So, $I=-\int \frac{dt}{t^{11}}=\frac{1}{10}\left(\frac{1}{t^{10}}\right)=\frac{1}{10(\ln (x+1)-\ln x{)}^{10}}+C$