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