`int(dx)/(x^(2)(x^(n)+1)^((n-1)/(n))`

Given that int (dx)/((1+ x ^(2))^n)=(x)/(2 (n-1)(1+x^(2) )^(n-1))+((2n -3))/(2(n-1))int (dx)/((1+ x ^(2))^(n-1)). Find the vlaue of int _(0)^(1) (dx)/((1+x ^(2) )^(4)), (you may or may not use reduction formula given)

If I_(n)=int cos^(n)x dx . Prove that I_(n)=(1)/(n)(cos^(n-1)x sinx)+((n-1)/(n))I_(n-2) .

The value of int(dx)/(x^n(1+x^n)^(1/ n)) is equal to

(d^n)/(dx^n)(logx)=? (a) ((n-1)!)/(x^n) (b) (n !)/(x^n) (c) ((n-2)!)/(x^n) (d) (-1)^(n-1)((n-1)!)/(x^n)

(d^n)/(dx^n)(logx)= ((n-1)!)/(x^n) (b) (n !)/(x^n) ((n-2)!)/(x^n) (d) (-1)^(n-1)((n-1)!)/(x^n)

The value of int_(a)^(b)(x^(n-1)((n-2)x^(2)+(n-1)(a+b)x+nab))/((x+a)^(2)(x+b)^(2))dx is equal to :

Let n in N such that n gt 1 . Statement-1: int_(oo)^(0) (1)/(1+x^(n))dx=int_(0)^(1) (1)/((1-x^(n))^(1//n))dx Statement-2: int_a^b f(x)dx=int_(a)^(b) f(a+b-x)dx

Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx . It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0 . int_(0)^(1)(x^(m-1)+x^(n-1))/((1+x)^(m+n))dx=

Prove that int_(0)^(1)(dx)/(1+x^(n))gt1-(1)/(n)"for n"inN

int_(2)^(4) (3x^(2)+1)/((x^(2)-1)^(3))dx = (lambda)/(n^(2)) where lambda, n in N and gcd(lambda,n) = 1 , then find the value of lambda + n