Concept:

Integral property:

• ∫ xⁿ dx = \(\rm x^{n+1}\over n+1\)+ C ; n ≠ -1
• \(\rm\int {1\over x} dx = \ln x\) + 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.

Calculation:

I = \(\rm \int {6x\over(3x^2+1)^2}\)

Let x² = t

Differentiating both sides

⇒ 2x dx = dt

Now,  I =  \(\rm \int {6x\over(3x^2+1)^2}\) dx

⇒ I = \(\rm 3\int {2x\over(3x^2+1)^2}\)dx

Substituting x² by t and dt = 2x dx

⇒ I = 3 \(\rm \int {1\over(3t+1)^2}\) dt

⇒ I = \(\rm 3\left[ {(3t+1)^{-1}\over-3}\right] + c\)

⇒ I = \(\rm {-1\over(3t+1)}\) + c

Substituting t as x²

⇒ I = \(\boldsymbol{\rm -{1\over3x^2+1}+c}\)