Question

Consider the following statements
Statement I The value of integral of the function $x(\log x{)}^2$ is $\frac{x^2}{2}(\log x{)}^2-\frac{x^2}{2}\log x+\frac{x^2}{4}+C$
Statement II The value of $\int \left(x^2+1\right)\log xdx$ is $\left(\frac{x^3}{3}+x\right)\log x+\frac{x^3}{9}+x+C$
Choose the correct option.

• A. Statement $I$ is true
• B. Statement II is true
• C. Both statements are true
• D. Both statements are false

Answer 1

Answer: (A)

I. Let $I=\int x(\log x{)}^{2}dx$
On taking $(\log x{)}^2$ as first function and $x$ as second function and integrating by parts, we get
$I=(\log x{)}^2\int xdx-\int \left[\frac{d}{dx}(\log x{)}^2\int xdx\right]dx$
$=(\log x{)}^2\cdot \frac{x^2}{2}-\int \left[\frac{2\log x}{x}\cdot \frac{x^2}{2}\right]dx$
$=\frac{x^2}{2}(\log x{)}^2-\int x\log xdx$
Again, integrating by parts, we get
$I=\frac{x^2}{2}(\log x{)}^2-\left[\log x\int xdx-\int \left\{\left(\frac{d}{dx}\log x\right)\int xdx\right\}dx\right]$
$=\frac{x^2}{2}(\log x{)}^2-\left[\frac{x^2}{2}\log x-\int \frac{1}{x}\cdot \frac{x^2}{2}dx\right]+C$
$=\frac{x^2}{2}(\log x{)}^2-\frac{x^2}{2}\log x+\frac{1}{2}\int xdx+C$
$=\frac{x^2}{2}(\log x{)}^2-\frac{x^2}{2}\log x+\frac{x^2}{4}+C$
II. Let $I=\int \left(x^2+1\right)\log xdx$
On taking log $x$ as first function and $\left(x^2+1\right)$ as second function and integrating by parts, we get
$I=\log x\int \left(x^2+1\right)dx-\int \left[\frac{d}{dx}(\log x)\int \left(x^2+1\right)dx\right]dx$
$\Rightarrow I=\log x\left(\frac{x^3}{3}+x\right)-\int \frac{1}{x}\cdot \left(\frac{x^3}{3}+x\right)dx$
$\Rightarrow I=\left(\frac{x^3}{3}+x\right)\log x-\int \left(\frac{x^2}{3}+1\right)dx$
$=\left(\frac{x^3}{3}+x\right)\log x-\frac{x^3}{9}-x+C$