`int (ax^2+bx+c)dx` =

If the two roots of the equation ax^2 + bx + c = 0 are distinct and real then

If sin theta and cos theta are the roots of the equation ax^2 - bx + c =0 , then a, b and c satisfy the relation

If int_0^3 (3ax^2+2bx+c)dx=int_1^3 (3ax^2+2bx+c)dx where a,b,c are constants then a+b+c=

If int e^(ax) sin bx dx=(e^(ax))/(sqrt(a^(2)+b^(2)))sin(bx+m)+c , then the value of m is-

If u=inte^(ax)cos bx dx and v=int e^(ax)sinbx dx , show that, (a^(2)+b^(2))(u^(2)+v^(2))=e^(2ax)

If u=inte^(ax)cos bx dx and v=int e^(ax)sinbx dx , show that, "tan"^(-1)(v)/(u)+"tan"^(-1)(b)/(a)=bx .

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 (xdx)/(sqrt(7x-10-x^(2))^3) can be evaluated by substituting for x as

Evaluate inte^(ax)sinbx dx and int e^(ax)cosbx dx by integrating each of them only once by parts.

Find Delta(Delta(ax^2 + bx + c)) , (a ne 0) .

If a, b, c are in G.P. and the equation ax^(2) + 2bx + c = 0 and dx^(2) + 2ex + f = 0 have a common root, then show that (d)/(a), (e)/(b), (f)/(c ) are in A.P.