(1)● To find the integral `color{red}{ ∫ (dx)/( ax^2 + bx +c)}` , we write
`=>`write it as ` color{green} {ax^2 + bx + c =a [ x^2 + b/a x + c/a ] }`
`= color{green} {a [(x+ b/(2a) )^2 + (c/a - b^2/(4a^2)) ]}`
`=>` Now, put `x + b/(2a) = t` so that dx = dt and writing ` color{blue} {c/a - b^2/(4a^2) = pm k^2}` We find the
integral reduced to the form `1/a ∫ (dt)/( t^2 pm k^2)` depending upon the sign of ` (c/a - b^2/(4a^2 ))` and hence can be evaluated.
(2)● To find the integral of the type `color{red}{ ∫ (dx)/sqrt(ax^2 + bx +c)}` , proceeding as in (1).
(3)● To find the integral of the type ` color{red}{∫ (px +q)/(ax^2 +bx +c)dx}` , where p, q, a, b, c are constants, we are to find real numbers A, B such that
`=>` `color {green }{px + q = A d/(dx) (ax^2 + bx+c) +B = A (2ax + b) +B}`
`=>` To determine A and B, we equate from both sides the coefficients of x and the constant terms. A and B are thus obtained and hence the integral is reduced to one of the known forms.
(4) ● For the evaluation of the integral of the type `color{green}{ ∫ ( (px+q)dx)/( sqrt (ax^2 +bx +c) ) }` , we proceed
as in (3) and transform the integral into known standard forms.
Let us illustrate the above methods by some examples.
(1)● To find the integral `color{red}{ ∫ (dx)/( ax^2 + bx +c)}` , we write
`=>`write it as ` color{green} {ax^2 + bx + c =a [ x^2 + b/a x + c/a ] }`
`= color{green} {a [(x+ b/(2a) )^2 + (c/a - b^2/(4a^2)) ]}`
`=>` Now, put `x + b/(2a) = t` so that dx = dt and writing ` color{blue} {c/a - b^2/(4a^2) = pm k^2}` We find the
integral reduced to the form `1/a ∫ (dt)/( t^2 pm k^2)` depending upon the sign of ` (c/a - b^2/(4a^2 ))` and hence can be evaluated.
(2)● To find the integral of the type `color{red}{ ∫ (dx)/sqrt(ax^2 + bx +c)}` , proceeding as in (1).
(3)● To find the integral of the type ` color{red}{∫ (px +q)/(ax^2 +bx +c)dx}` , where p, q, a, b, c are constants, we are to find real numbers A, B such that
`=>` `color {green }{px + q = A d/(dx) (ax^2 + bx+c) +B = A (2ax + b) +B}`
`=>` To determine A and B, we equate from both sides the coefficients of x and the constant terms. A and B are thus obtained and hence the integral is reduced to one of the known forms.
(4) ● For the evaluation of the integral of the type `color{green}{ ∫ ( (px+q)dx)/( sqrt (ax^2 +bx +c) ) }` , we proceed
as in (3) and transform the integral into known standard forms.
Let us illustrate the above methods by some examples.