\( \int \frac { \sqrt { x } } { 1 + \sqrt[ 4 ] { x } } d x \)

To evaluate the integral \[ \int \frac{\sqrt{x}}{1 + \sqrt[4]{x}} \, dx, \] let's proceed with a substitution that simplifies the integral. **Step 1: Substitution** Let \( t = \sqrt[4]{x} = x^{1/4} \). Then: \[ x = t^4 \quad \text{and} \quad dx = 4t^3 \, dt. \] Also, \( \sqrt{x} = x^{1/2} = (t^4)^{1/2} = t^2 \). **Step 2: Rewrite the Integral** Substituting these into the original integral: \[ \int \frac{t^2}{1 + t} \cdot 4t^3 \, dt = 4 \int \frac{t^5}{1 + t} \, dt. \] **Step 3: Polynomial Division** To integrate \( \frac{t^5}{1 + t} \), perform polynomial division: \[ \frac{t^5}{1 + t} = t^4 - t^3 + t^2 - t + 1 - \frac{1}{1 + t}. \] Thus, the integral becomes: \[ 4 \int \left( t^4 - t^3 + t^2 - t + 1 - \frac{1}{1 + t} \right) dt. \] **Step 4: Integrate Term by Term** Integrate each term separately: \[ 4 \left( \frac{t^5}{5} - \frac{t^4}{4} + \frac{t^3}{3} - \frac{t^2}{2} + t \right) - 4 \ln|1 + t| + C, \] where \( C \) is the constant of integration. **Step 5: Substitute Back** Recall that \( t = x^{1/4} \). Substituting back: \[ \frac{4}{5} x^{5/4} - x + \frac{4}{3} x^{3/4} - 2x^{1/2} + 4x^{1/4} - 4 \ln\left(1 + \sqrt[4]{x}\right) + C. \] **Final Answer:** \[ \int \frac{\sqrt{x}}{1 + \sqrt[4]{x}} \, dx = \frac{4}{5}x^{5/4} - x + \frac{4}{3}x^{3/4} - 2\,x^{1/2} + 4\,x^{1/4} - 4\,\ln\left(1 + \sqrt[4]{x}\right) + C, \] where \( C \) is the constant of integration.