Concept:

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 be determined.
• Substitution is done and integral function is then integrated.
• Finally, initial variable t, to be returned.

Calculation:

I = \(\rm \int {6\over(\sin x +\sin2x)}\) dx

Using integration by parts

⇒ I = \(\rm \int {6\over(\sin x +2\sin x \cos x)}\) dx

⇒ I = \(\rm \int {6\over\sin x(1 +2 \cos x)}\) dx

Multiply sin x both in numerator and denominator

⇒ I = \(\rm \int {6\sin x\over\sin^2 x(1 +2 \cos x)}\) dx

⇒ I = \(\rm \int {6\sin x\over(1-\cos x)(1+\cos x)(1 +2 \cos x)}\) dx

Substituting cos x = t ⇒ -sin x dx = dt

⇒ I = \(\rm \int {-6\over(1-t)(1+t)(1+2t)}\) dt

By partial fraction

⇒ I = \(\rm -6 \int {1\over6(1-t)}-{1\over2(1+t)}+{4\over3(1+2t)}\) dt

⇒ I = \(\rm -6[ {-1\over6}\ln|(1-t)|-{1\over2}\ln|(1+t)|+{4\over3\times2}\ln|(1+2t)|]\) + C

⇒ I = \(\rm \ln|(1-t)|+3\ln|(1+t)|-4\ln|(1+2t)|\) + C

⇒ I = \(\boldsymbol{\rm \ln|1-\cos x|+3\ln|1+\cos x|-4\ln|1+2\cos x|}\) + 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.

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