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 = ∫ (x2 cos x) dx

Using integration by parts

⇒ I = x² ∫ cos x dx - ∫ \(\rm \left({dx^2 \over dx}\times ∫ \cos xdx\right)\) dx + C

⇒ I = x² sin x - ∫\(\rm \left(2x\sin x\right)\) dx + C

⇒ I = x2 sin x - 2 ∫ \(\rm x\sin x\) dx + C

⇒ I = x2 sin x - 2[x ∫ sin x dx - ∫ \(\rm \left({dx \over dx}\times ∫ \sin xdx\right)\)dx] + C

⇒ I = x2 sin x - 2x (-cos x) + 2 ∫ - cos x dx + C

⇒ I = x2 sin x + 2x cos x - 2 sin x + C