If `b=int_(0)^(1) (e^(t))/(t+1)dt`, then `int_(a-1)^(a) (e^(-t))/(t-a-1)` is

If int_(0)^(1)(e^(t))/(1+t)dt=a then find the value of int_(0)^(1)(e^(t))/((1+t)^(2))dt in terms of a .

Ifint_0^1(e^t dt)/(t+1)=a ,then find the value of int_(b-1)^b(e^(-t)dt)/(t-b-1)

If lambda=int_0^1(e^t)/(1+t) , then int_0^1e^tlog_e(1+t)dt is equal to

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If f(x)=x+int_(0)^(1)t(x+t)f(t)dt , then the value of (23)/(3)f(0) is equal to-

If f(t) is an odd function, then int_(0)^(x)f(t) dt is -

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

If f(x)=int_(x^2)^(x^2+1)e^(-t^2)dt , then f(x) increases in

Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., x=x(t), y=(t), then the area of the figure is evaluated by one of the three formulas : S=-int_(alpha)^(beta) y(t)x'(t)dt, S=int_(alpha)^(beta) x(t)y'(t)dt, S=(1)/(2)int_(alpha)^(beta)(xy'-yx')dt, Where alpha and beta are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t. If the curve given by parametric equation x=t-t^(3), y=1-t^(4) forms a loop for all values of t in [-1,1] then the area of the loop is

If int_(0)^(x)f(t)dt=x^2+int_(x)^(1)t^2f(t)dt , then f((1)/(2)) is equal to