If `I_n=intdx/(x^2+a^2)^n,ninN`, then show that: `I_(n+1)=1/(2na^2)x/((x^2+a^2)^n)+(2n-1)/(2n). 1/a^2I_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(a^(2)+x^(2))^((n)/(2))dx, then show that I_(n)=((x(a^(2)+x^(2))^((n)/(2)))/(n+1))+(na^(2))/(n+1)I(n-2)

If I_(n)=int(x^(n))/(1+x^(2))dx, where n in N , then : I_(n+2)+I_(n)=

If n is an integer then show that (1+i)^(2n)+(1-i)^(2n)=2^(n+1)cos(n pi)/(2)

If I_(n)=int(x^(n)dx)/(sqrt(x^(2)+a)) then prove that I_(n)+(n-1)/(n)al_(n-2)=(1)/(n)x^(n-1)*sqrt(x^(2)+a)

If I_n=intsqrt((a^2+x^2)^n)dx , Prove that: I_n=(xsqrt((a^2+x^2))^n)/(n+1)+(na^2)/(n+1)int(a^2+x^2)^(n/2-1)dx

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"):}

Prove that: I_(n)=int_(0)^(oo)x^(2n+1)e^(-x^(2))dx=(n!)/(2),n in N

If I_(n)=int_(0)^(1)x^(n)(tan^(-1)x)dx, then prove that(n+1)I_(n)+(n-1)I_(n-2)=-(1)/(n)+(pi)/(2)

Show that: (x)+(x+(1)/(n))+(x+(2)/(n))+...+(x+(n-1)/(n))=nx+(n-1)/(2)