Question

If $I_{m,n}=\int x^m\cdot (\ln x{)}^{n}dx,$ and $I_{10,1}=\frac{x^k\ln x}{k}-\frac{x^k}{p}+C,$ then which of the following is/are true
(where $C$ is integration constant)

• A. $k=11$
• B. $k=p^2$
• C. $k^2=p$
• D. $p=100$

Answer 1

Answer: (A), (C)

$k^2=p$

$I_{10,1}=\int x^{10}\cdot \ln xdx$
$=\int \ln x\cdot x^{10}dx$
$=\ln x\cdot \frac{x^{11}}{11}-\int \frac{x^{11}}{11}\cdot \frac{1}{x}dx=\frac{x^{11}\cdot \ln x}{11}-\frac{1}{11}\cdot \int x^{10}dx=\frac{x^{11}\cdot \ln x}{11}-\frac{x^{11}}{121}+C$
So, $k=11$ and $p=121$