Make a substitution to convert the integrand to a rational f

Make a substitution to convert the integrand to a rational function and hence find the integral integral dx/x^2 + x Squareroot x

Solution

We have 1/(x²+xx) = (x² -xx)/( x²+xx)( x²-xx) = (x²-xx)/(x⁴–x³) = (x -x)/x²(x-1) = [1/x(x-1)] – [1/x^3/2(x-1)]. Then, dx/(x² + xx) = dx/x(x-1) - dx/x^3/2(x-1).

Further, 1/x(x -1) = [1/(x-1) -1/x] so that dx/x(x-1) = dx/(x-1) - dx/x = log(x-1) – logx …(1)

Let us now determine dx/x^3/2(x-1) by substituting x = u . Then x = u² and dx = 2udu. Also, dx/x^3/2(x-1) = 2udu/u³(u²-1) = 2du/u²(u² -1) = 2du[1/(u² -1) -1/u²] = 2du/(u² -1)   - 2du/u² . Now 2du/u² = -2/u +c (where c is a constant) = - 2/x +c…(2)

Further 2du/(u² -1)   = 2du/(u-1)(u+1) = du/ [ 1/(u -1)-1/(u+1)] = du/(u-1) - du/(u+1) = log(u-1) –log(u+1) = log(x -1) –log(x +1).

Finally, dx/(x² + xx) = log(x-1) – logx - log(x -1) +log(x +1) - 2/x +c where c is a constant.