`int_(0)^(2)x(2-x)^(n)dx`

Evaluate : (i) int_(0)^(1)x(1-x)^(n)dx (ii) int_(0)^(1)x(1-x)^(3//2)dx

int_(0)^(2)[x]dx

int_(0)^(2)[x]dx

evaluate int_(0)^(1)x^(2)(1-x)^(n)dx

int_(0)^(2)(x^(2)+x)dx

Evaluate int_(0)^(2pi)(xcos^(2n)x)/(cos^(2n)x+sin^(2n)x)dx

Prove that: int_(0)^(2a)f(x)dx=int_(0)^(2a)f(2a-x)dx

lim_(n rarr oo)(n)/(2^(n))int_(0)^(2)x^(n)dx equals

If int_(0)^(1)f(x)dx=1, int_(0)^(1)x f(x)dx=a and int_(0)^(1)x^(2)f(x)dx=a^(2) , then : int_(0)^(1)(a-x)^(2)f(x)dx=

show that (a) int_(0) ^(2pi) sin ^(3) x dx = 0 , (b) int_(-1)^(1) e^(-x^(2)) dx = 2 int_(0)^(1) e^(-x^(2)) dx