Homework Sourseintegrate 4x2912x4Solution integral sqrt94 x2x4 dx Then sqrt
integrate (4x^2+9)^(1/2)/x^4
Solution
integral sqrt(9+4 x^2)/x^4 dx Then sqrt(4 x^2+9) = sqrt(9 tan^2(u)+9) = 3 sec(u) and u = tan^(-1)((2 x)/3): = 9/2 integral 16/81 cot(u) csc^3(u) du = 8/9 integral cot(u) csc^3(u) du For the integrand cot(u) csc^3(u), substitute s = csc(u) and ds = -cot(u) csc(u) du: = -8/9 integral s^2 ds = -(8 s^3)/27+constant = -8/27 csc^3(u)+constant = -((4 x^2)/9+1)^(3/2)/x^3+constant = -(4 x^2+9)^(3/2)/(27 x^3)+constant
