Evaluate: If `intcos^n xdxdotp rov et h a t I_n=1/n(cos^(n-1)xsinx)+((n-1)/x)I_(n-2)`

If l_(n)=intcos^(n)xdx, prove that l_(n)=(1)/(n)(cos^(n-1)xsinx)+((n-1)/(n))l_(n-2) .

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) .

If I_(n)=int cos^(n)xdx, then (n+1)I_(n+1)-nI_(n-1)=

If I_(m"," n)=int cos^(m)x*cos nx dx , show that (m+n)I_(m","n)=cos^(m)x*sin nx+m I_((m-1","n-1))

If l_(n)=int e^(mx)cos^(n)xdx then prove that (m^(2)+n^(2))I_(n)=e^(mx)*(m cos x+n sin x)cos^(n-1)x+n(n-1)l_(n)

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"):}

If I_(n)=int x^(n)cos axdx and I_(n)=int x^(n)sin axdx then show that.(1)aI_(n)=x^(n)sin ax-nI_(n-1)

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_(0)^( pi/4)tan^(n)xdx, prove that I_(n)+I_(n-2)=(1)/(n+1)

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)