`sqrt(ax^2+bx+c)`

Euler's substitution: Integrals of the form intR(x, sqrt(ax^(2)+bx+c))dx are claculated with the aid of one of the following three Euler substitutions: i. sqrt(ax^(2)+bx+c)=t+-x sqrt(a)if a gt 0 ii. sqrt(ax^(2)+bx+c)=tx+-x sqrt(c)if c gt 0 iii. sqrt(ax^(2)+bx+c)=(x-a)t if ax^(2)+bx+c=a(x-a)(x-b) i.e., if alpha is real root of ax^(2)+bx+c=0 int(xdx)/((sqrt(7x-10-x^(2)))^(3)) can be evaluated by substituting for x as

Consider the following expressions : 1. x+x^(2)- 1/x 2. sqrt(ax^(2) + bx + x - c + d/c - e / x^(2)) 3. 3x^(2) - 5 x + ab 5 1/x - 2/ (x+5) Which of the above are rational expressions ?

Differentiate each of the following w.r.t. x: (i) (ax+b)^(m)" "(ii)(2x+3)^(5)" "(iii)sqrt(ax^(2)+2bx+c)

Differentiate : (i) (ax + b)^(m) , (ii) (3x+5)^(6) , (iii) sqrt(ax^(2) + 2bx + c)

If lim_(x rarr oo)ax+sqrt(x^(2)+bx)=3

(ax^(2) + bx + c)^(n)

If a+b+c=0 , then find the value of sqrt(x^a.x^b.x^c) .

int sqrt(x)(ax^(2)+bx+c)dx