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:

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

Using integration by parts

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

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

⇒ I = \(\rm {x^2\over2}\) ln x - ∫ \(\rm {x\over2}\) dx + C

⇒ I = \(\boldsymbol{\rm {x^2\ln x\over2}-}\) \(\boldsymbol{\rm {x^2\over4}}\) + C