Solve $\displaystyle\displaystyle \int{x\ln x}\mathrm{d}x$.

Techniques of  Integration

H2 Math

International Baccalaureate Higher Level Mathematics

IB HL Math

Worked Example 4

**LIATE - Preference for choosing $u$:**
• L - Logarithmic functions ($\ln x$, $\log x$, ...)
• I - Inverse trigonometric functions ($\tan^{-1}x$, $\sin^{-1}x$, $\cos^{-1}x$)
• A - Algebraic functions ($x^2$, $x^3$, $2x^9$, ...)
• T - Trigonometric functions ($\cos x$, $\sin x$, $\tan x$)
• E - Exponential functions ($e^x$, $2^x$, ...)

$\displaystyle\int x \ln x \, \mathrm{d}x$

Let $u = \ln x$, $\displaystyle\frac{\mathrm{d}v}{\mathrm{d}x} = x$
$\displaystyle\frac{\mathrm{d}u}{\mathrm{d}x} = \frac{1}{x}$, $v = \displaystyle\frac{x^2}{2}$

$= uv - \displaystyle\int v \frac{\mathrm{d}u}{\mathrm{d}x} \, \mathrm{d}x$
$= (\ln x)\left(\displaystyle\frac{x^2}{2}\right) - \displaystyle\int \frac{x^2}{2} \cdot \frac{1}{x} \, \mathrm{d}x$

$= \displaystyle\frac{1}{2}x^2 \ln|x| - \frac{1}{2}\displaystyle\int x \, \mathrm{d}x$

$= \displaystyle\frac{1}{2}x^2 \ln|x| - \frac{1}{2}\left(\frac{x^2}{2}\right) + c$

$= \displaystyle\frac{1}{2}x^2 \ln|x| - \frac{1}{4}x^2 + c$

Worked Example 4

Worked Example 4

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