`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(m,n)=int_0^1x^(m-1)(1-x)^(n-1)dx , then

Let I_(n) = int_(0)^(1)x^(n)(tan^(1)x)dx, n in N , then

If I_n=int_0^(pi//4)tan^("n")x dx , prove that I_n+I_(n-2)=1/(n-1)dot

If I_n=int_0^ooe^(-x)x^(n-1)log_exdx , then prove that I_(n+2)-(2n+1)I_(n+1)+n^2I_n=0

IfI_n=int_0^pix^nsinx dx ,t h e n fin d t h e v a l u eof I_5+20 I_3dot

If I_(n)=int_(0)^(pi/2) sin^(n)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"):}

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

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

If I_(n)=int_(0)^(pi//4) tan^(n)x dx, (ngt1 is an integer ), then (a) I_(n)+I_(n-2)=1/(n+1) (b) I_(n)+I_(n-2)=1/(n-1) (c) I_(2)+I_(4),I_(6),……. are in H.P. (d) 1/(2(n+1))ltI_(n)lt1/(2(n-1))

Iff(x)=int_1^x(logt)/(1+t+t^2)dxAAxlt=1,t h e np rov et h a tf(x)=f(1/x)dot