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

Substitution method: If the function cannot be integrated directly substitution method is used. To integration by substitution is used in the following steps:

• A new variable is to be chosen, say “t”
• The value of dt is to is to be determined.
• Substitution is done and integral function is then integrated.
• Finally, initial variable t, to be returned.

I = \(\rm ∫ {\sin (x+a)\over \sin x}\) dx

⇒ I = \(\rm ∫ {(\sin x\cos a + \cos x \sin a)\over \sin x} dx\)

⇒ I = \(\rm ∫ {(\sin x\cos a)\over\sin x} + {(\cos x \sin a)\over \sin x} dx\)

⇒ I = ∫ cos a dx + sin a ∫ \(\rm \cos x\over \sin x\) dx

Let sin x = t ⇒ cos x dx = dt 

Substituting sin x as t

⇒ I = x cos a + sin a ∫ \(\rm dt\over t\) + C

⇒ I = x cos a + sin a [ln(|t|)] + C

⇒ I = x cos a + sin a ln(|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.