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 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)=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)

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

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) .

int(2x^(n-1))/(x^(n)+3)dx

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

Prove that: int_(0)^(2 pi)(x sin^(2n)x)/(sin^(2n)+cos^(2n)x)dx=pi^(2)