" 8."int_(0)^(1)x^(2)(1-x^(2))^(5/2)dx=

int_(0)^(1)x^(2)(1-x)^(5)dx=

int_(0)^(1)x^(4)(1-x)^(5//2)dx=

int_(0)^(1)x^(2)sqrt(1-x^(2))dx=

int_(0)^(1)(2-x^(2))/(1+x^(2))dx=

" 2."int_(0)^(1)(x^2)/(1+x^(3))dx

int_(0)^(1) x^(2)/(x^(2)+1)dx=

If m gt 0, n gt 0 , the definite integral l=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx depends upon the vlaues of m and n and is denoted by beta(m,n) , called the beta function. E.g. int_(0)^(1)x^(4)(1-x)^(5)dx=int_(0)^(1)x^(5-1)(1-x)^(6-1)dx=beta(5, 6) and int_(0)^(1)x^(5//2)(1-x)^(-1//2)dx=int_(0)^(1)x^(7//2-1)(1-x)^(1//2-1)dx=beta((7)/(2),(1)/(2)) . Obviously, beta(n, m)=beta(m, n) . If int_(0)^(n)(1-(x)/(n))^(n)x^(k-1)dx=R beta(k, n+1) , then R is equal to

int_(0)^(2)(x^(2)+1)dx

int_(0)^(1)(2x)/((1+x^(2)))dx

int_(0)^(1)(3x^(2)+5x)dx