`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))

If I_(n)=int_(0)^(pi//4) tan^(n)x dx, (ngt1 is an integer ), then

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

If I_n=int( lnx)^n dx then I_n+nI_(n-1)

Let I_(n)=int_(0)^(pi//2)(sinx+cosx)^(n)dx(nge2) . Then the value of n. I_(n)-2(n-1)I_(n-1) is

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

Let I_(n)=int_(0)^(1)x^(n)sqrt(1-x^(2))dx. Then lim_(nrarroo)(I_(n))/(I_(n-2))=

If n is a positive integer, prove that |I m(z^n)|lt=n|Im(z)||z|^(n-1.)

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))

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_0^1(1-x^5)^n dx ,t h e n(55)/7(I_(10))/(I_(11))i se q u a lto___