If `I_n=int x^nsqrt(a^2-x^2)dx,` prove that `I_n=-(x^(n-1)(a^2-x^2)^(3/2))/((n+2))+((n+1))/((n+2))a^2I_(n-2)`

If I_(n)=int x^(n)sqrt(a^(2)-x^(2))dx, prove that I_(n)=-(x^(n-1)(a^(2)-x^(2))^((3)/(2)))/((n+2))+((n+1))/((n+2))a^(2)I_(n-2)

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=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=intxcos^nxdx,prove that: I_n=1/n cos^(n-1)x(sinx)+(n-1)/(n)I_(n-2)

If I_(n) = int (sin nx)/(cosx)dx , prove that I_(n)=-(2)/(n-1)cos(n-1)x-I_(n-2)

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)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)=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)

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

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