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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).
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