Evaluate \(\rm \displaystyle\int x \log x \ dx\)

Concept:

Integral property:

• ∫ xn dx = \(\rm x^{n+1}\over n+1\)+ C ; n ≠ -1
• \(\rm∫ {1\over x} dx = \ln x\) + C
• ∫ ex dx = ex+ C
• ∫ ax dx = (ax/ln a) + C ; a > 0,  a ≠ 1
• ∫ sin x dx = - cos x + C
• ∫ cos x dx = sin x + C

Integration by parts: Integration by parts is a method to find integrals of products. The formula for integrating by parts is given by:

⇒ \(\rm ∫ u vdx=u ∫ vdx- ∫ \left({du\over dx}\times ∫ vdx\right)dx \) + C

where u is the function u(x) and v is the function v(x) 

ILATE rule is Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric, and Exponent.

Calculation:

I = \(\rm \int xlog xdx \)

I = \(\rm \ln x∫ xdx- ∫ \left({d\over dx}(\log x)\times ∫ xdx\right)dx +c\)

I = \(\rm {x^2\over2}\log x- ∫ \left({1\over x}\times{x^2\over2}\right)dx +c\)

I = \(\rm {x^2\over2}\log x- ∫ \left({x\over2}\right)dx +c\)

I = \(\rm {x^2\over2}\log x-{x^2\over4} +c\)