Evaluate for `m in N ,` `int(x^(3m)+x^(2m)+x^m)(2x^(2m)+3x^m+6)^(1/m)dx ,x >0`

If m is a non-zero number and int (x^(5m-1)+2x^(4m-1))/(x^(2m)+x^m+1)^3 dx=f(x)+c , then f(x) is:

Integrate : int (e^(m tan^(-1)x))/((1+x^(2))^((3)/(2)))dx

lim_(xto0) ((2^(m)+x)^(1//m)-(2^(n)+x)^(1//n))/(x) is equal to

Evaluate : int ((x-l)(x-m))/((x-a)(x-b))dx

Find the reduction for I_m,n = int (sin x)^m (cos x)^n dx

Evaluate: int[(x-l)(x-m)]/[(x-a)(x-b)]dx

IfI(m , n)=int_0^1x^(m-1)(1-x)^(n-1)dx ,(m , n in I ,m ,ngeq0),t h e n (a) I(m , n)=int_0^oo(x^(m-1))/((1+x)^(m-n))dx (b) I(m , n)=int_0^oo(x^(m-1))/((1+x)^(m+n))dx (c) I(m , n)=int_0^oo(x^(n-1))/((1+x)^(m+n))dx (d) I(m , n)=int_0^oo(x^n)/((1+x)^(m+n))dx

The number of integral values of m for which the quadratic expression (1 + 2m)x^(2) - 2(1 + 3m)x + 4(1 + m), x in R , is always positive is

Prove that : ((x^m)/(x^n))^(m+n-l) xx((x^n)/(x^l))^(n+l-m)xx((x^l)/(x^m))^(l+m-n) =1 .