`"If " (d)/(dx)f(x)=f'(x), " then " int(xf'(x)-2f(x))/(sqrt(x^(4)f(x)))dx ` is equal to

(d)/(dx)(int_(f(x))^(g(x)) phi(t)dt) is equal to

If d/dx (ϕ(x))=f(x) , then int_(1)^(2) f(x) dx is ______

Let f be a differentiable function satisfying f(xy)=f(x).f(y).AA x gt 0, y gt 0 and f(1+x)=1+x{1+g(x)} , where lim_(x to 0)g(x)=0 then int (f(x))/(f'(x))dx is equal to

Let f (x) be a twice differentiable function defined on (-oo,oo) such that f (x) =f (2-x)and f '((1)/(2 )) =f' ((1)/(4))=0. Then int _(-1) ^(1) f'(1+ x ) x ^(2) e ^(x ^(2))dx is equal to :

If (d)/(dx)f(x)=g(x) for a le x le b then, overset(b)underset(a)int f(x) g(x) dx equals

If f(x)=(x^2)/(|x|) , write d/(dx)(f(x))

Statement 1: If differentiable function f(x) satisfies the relation f(x)+f(x-2)=0AAx in R , and if (d/(dx)f(x))_(x=a)=b ,t h e n(d/(dx)f(x))_(x=a+4000)=b . Statement 2: f(x) is a periodic function with period 4.

Let f(x) is differentiable function satisfying 2int_(1)^(2)f(tx) dt = x+2, AA x in R Then int_(1)^(2)(8f(8x)-f(x)-21x) dx equal to

The integral inte^x(f(x)+f\'(x))dx can be solved by using integration by parts such that: I=inte^xf(x)dx+inte^xf\'(x)dx=e^xf(x)-inte^xf\'(x)dx+inte^xf\'(x)dx=e^xf(x)+C , and inte^(ax)(f(x)+(f\'(x))/a)dx=e^(ax)f(x)/a+C ,Now answer the question: inte^x x^x(2+logx)= (A) e^x x^xlogx+C (B) e^x+x^x+C (C) e^x x(logx)^2+C (D) e^x.x^x+C

According to Leibritz differentiation under the sign of integration can be performed as as below. (i) (d)/(dx)[int_(phi(x))^(Psi(x))f(t)dt]=f{Psi(x)}xx(d)/(dx){Psi(x)}-f{phi(x)}xx(d)/(dx){phi(x)} (ii) (d)/(dx)[int_(phi(x))^(Psi(x))f(x,t)dt]=int_(phi(x))^(Psi(x))(del)/(delx)(f(x,t)dt)+f(x,Psi(x))xx(d)/(dx)Psi(x)-f(x,phi(x))xx(d)/(dx)(phi(x)) int_(x^(2))^(x^(3))cost^(2)dt has the derivative