If `g(x)=(f(x))/((x-a)(x-b)(x-c))`,where f(x) is a polynomial of degree `<3` , then `intg(x)dx=|[1,a,f(a)log|x-a|],[1,b,f(b)log|x-b|],[1,c,f(c)log|x-c|]|-:|[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|+k` `(dg(x))/(dx)=|[1,a,-f(a)(x-a)^(-2)],[1,b,-f(b)(x-b)^(-2)],[1,c,-f(c)(x-c)^(-2)]|:-|[1,a,a^2],[1,b,b^2],[1,c,c^2]|`

If int(tan^(9)x)dx=f(x)+log|cosx|, where f(x) is a polynomial of degree n in tan x, then the value of n is

If int ((2x+3)dx)/(x(x+1)(x+2)(x+3)+1)=C-1/(f(x)) where f(x) is of the form of ax^(2)+bx+c then (a+b+c) equals

For every function f (x) which is twice differentiable , these will be good approximation of int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)} , for more acutare results for cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)] When c= (a+b)/(2) int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx If lim_(t toa) (int_(a)^(t)f(x)dx-((t-a))/(2){f(t)+f(a)})/((t-a)^(3))=0 , then degree of polynomial function f (x) atmost is

Let alpha be a repeated root of a quadratic equation f(x)=0a n dA(x),B(x),C(x) be polynomials of degrees 3, 4, and 5, respectively, then show that |A(x)B(x)C(x)A(alpha)B(alpha)C(alpha)A '(alpha)B '(alpha)C '(alpha)| is divisible by f(x) , where prime (') denotes the derivatives.

If int((2x+3)dx)/(x(x+1)(x+2)(x+3)+1)=C-(1)/(f(x)) where f(x) is of the form of ax^(2)+bx+c , then the value of f(1) is

If f(x),g(x)a n dh(x) are three polynomial of degree 2, then prove that varphi(x)=|f(x)g(x)h(x)f'(x)g'(x h '(x)f' '(x)g' '(x h ' '(x)| is a constant polynomial.

If f(x),g(x)a n dh(x) are three polynomials of degree 2, then prove that varphi(x)=|f(x)g(x)h(x)f^(prime)(x)g^(prime)(x)h^(prime)(x)f^(x)g^(x)h^(x)|i sacon s t a n tpol y nom i a l

Solve (dy)/(dx) + yf^(')(x) = f(x) f^(')(x) , where f(x) is a given integrable function of x .

Let f(x)=|x-1|dot Then (a) f(x^2)=(f(x))^2 (b) f(x+y)=f(x)+f(y) (c) f(|x|)-|f(x)| (d) none of these

If f(x), h(x) are polynomials of degree 4 and |(f(x), g(x),h(x)),(a, b, c),(p,q,r)| =mx^4+nx^3+rx^2+sx+r be an identity in x, then |(f''(0) - f''(0),g''(0) - g''(0),h''(0) -h''(0)),(a,b,c),(p,q,r)| is