Let `f(x) = (1 - x)^2 sin^2 x + x^2 ` for all x ∈ R, and let `g(x) = ∫((2(t - 1))/(t + 1) - ln t)f(t)dt` for t ∈ [1, x] for all x ∈ (1, ∞).Which of the following is true ?

Let f(x) = (1-x)^(2) sin^(2)x+ x^(2) for all x in IR and let g(x) = int_(1)^(x)((2(t-1))/(t+1)-lnt) f(t) dt for all x in (1,oo) . Which of the following is true ?

Let f(x) = (1-x)^(2) sin^(2)x+ x^(2) for all x in IR and let g(x) = int_(1)^(x)((2(t-1))/(t+1)-lnt) f(t) dt for all x in (1,oo) . Consider the statements : P : There exists some x in IR such that f(x) + 2x = 2 (1+x^(2)) Q : There exist some x in IR such that 2f(x) + 1 = 2x(1+x) Then

For all t gt 0 , f(x) = (t^(2)-1)/(t-1)-t . Which of the following is true about f(t) ?

Let g(x)=int_0^(e^x) (f\'(t)dt)/(1+t^2) . Which of the following is true?

Let f(x) = int_(-2)^(x)|t + 1|dt , then

Let f (x) =(1)/(x ^(2)) int _(4)^(x) (4t ^(2) -2 f '(t) dt, find 9f'(4)

Let f (x)= int _(x^(2))^(x ^(3))(dt)/(ln t) for x gt 1 and g (x) = int _(1) ^(x) (2t ^(2) -lnt ) f(t) dt(x gt 1), then: (a) g is increasing on (1,oo) (b) g is decreasing on (1,oo) (c) g is increasing on (1,2) and decreasing on (2,oo) (d) g is decreasing on (1,2) and increasing on (2,oo)

Let f(x)=int_(1)^(x)(3^(t))/(1+t^(2))dt , where xgt0 , Then

If int_(0) ^(x) f (t) dt = x + int _(x ) ^(1) t f (t) dt, then the value of f (1) , is

f(x) : [0, 5] → R, F(x) = int_(0)^(x) x^2 g(x) , f(1) = 3 g(x) = int_(1)^(x) f(t) dt then correct choice is