how to integrate ∫√(linear)/quadratic dx

let the integral be (ax+ b)^0.5/(px^2+qx+r)

substitute ax+b =1/t^2

then the integral will be reduced to the form dx/(ax^4+bx^2+c) which can be easily integrated by dividing num & deno by ax^2

Hi Suchita,

This is an integral which can be integrated by the substitution

linear = t².

Say it is ∫√(ax+b)/(px²+qx+r).

So make the substitution, and you have an integral of the form ∫At²/(A₁t⁴+A₂t²+A₃)----- A, A1, A2, A3 are real constants.

So for the denominator, split it into two factors A₁(t²+a)(t²-a) {Here note that, "a" can be real or complex number which is a constant, depending of the descriminant of the equation in the denominator).

According to the factors in the Dr, write the Nr as C*{(t²+a)+(t²-a)}, where C is a constant.

Now split it into two integrals, and you can will have integrals of the form 1/(x^2+a^2) and 1/(x^2-a^2), for which we have standard formulas, in terms of inverse tan and log respectively.

Unfortunately this method is not mentioned in any book, as a standard integration. But now you know how to do it.

Hope that helps.

All the best.

Regards,

Ashwin (IIT Madras).