If `I_n=int_0^1(dx)/((1+x^2)^n); n in N ,` then prove that `2nI_(n+1)=2^(-n)+(2n-1)I_n`

Prove that ((n + 1)/(2))^(n) gt n!

If I_(n)=int_(0)^(pi/2) sin^(x)x dx , then show that I_(n)=((n-1)n)I_(n-2) . Hence prove that I_(n)={(((n-1)/n)((n-3)/(n-2))((n-5)/(n-4))………(1/2)(pi)/2,"if",n"is even"),(((n-1)/n)((n-3)/(n-2))((n-5)/(n-4))………(2/3)1,"if",n"is odd"):}

IfI_n=int_0^1x^n(tan^(-1)x)dx ,t h e np rov et h a t (n+1)I_n+(n-1)I_(n-2)=-1/n+pi/2

IfI_n=int_0^1(dx)/((1+x^2)^n),w h e r en in N , which of the following statements hold good? 2nI_(n+1)=2^(-n)+(2n-1)I_n I_2=pi/8+1/4 (c) I_2=pi/8-1/4 I_3=(3pi)/(32)+1/4

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_m-2n(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))

Prove that n! ( n +2) = n ! + ( n + 1) !

Prove that ((2n)!)/(n!) =2^(n) (1,3,5,……..(2n-1)) .

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) .

IfI_n=int_0^(pi/4)tan^n xdx ,(n >1 and is an integer), then I_n+I_(n-2)=1/(n+1) I_n+I_(n-2)=1/(n-1) I_2+I_4,I_4+I_6, ,a r einHdotPdot 1/(2(n+1))