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

To evaluate the integral: \[ \int \frac{\sqrt{x}}{1 + \sqrt[4]{x}} \, dx \] we can use a substitution method to simplify the expression. ### Step 1: Substitution Let: \[ t = \sqrt[4]{x} \quad \text{which means} \quad x = t^4 \] Then, differentiate both sides with respect to \( t \): \[ dx = 4t^3 \, dt \] Also, note that: \[ \sqrt{x} = x^{1/2} = (t^4)^{1/2} = t^2 \] ### Step 2: Rewrite the Integral in Terms of \( t \) Substituting \( x = t^4 \) and \( dx = 4t^3 \, dt \): \[ \int \frac{\sqrt{x}}{1 + \sqrt[4]{x}} \, dx = \int \frac{t^2}{1 + t} \cdot 4t^3 \, dt = 4 \int \frac{t^5}{1 + t} \, dt \] ### Step 3: Simplify the Integrand We can perform polynomial division on \( \frac{t^5}{1 + t} \): \[ \frac{t^5}{1 + t} = t^4 - t^3 + t^2 - t + 1 - \frac{1}{1 + t} \] 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: \[ 4 \left( \frac{t^5}{5} - \frac{t^4}{4} + \frac{t^3}{3} - \frac{t^2}{2} + t - \ln|1 + t| \right) + C \] ### Step 5: Substitute Back in Terms of \( x \) Recall that \( t = \sqrt[4]{x} \), so: \[ t^5 = x^{5/4}, \quad t^4 = x, \quad t^3 = x^{3/4}, \quad t^2 = x^{1/2}, \quad t = x^{1/4} \] Substituting back: \[ 4 \left( \frac{x^{5/4}}{5} - \frac{x}{4} + \frac{x^{3/4}}{3} - \frac{x^{1/2}}{2} + x^{1/4} - \ln(1 + x^{1/4}) \right) + C \] ### Final Answer Combining the constants, the final expression is: \[ \int \frac{\sqrt{x}}{1 + \sqrt[4]{x}} \, dx = \frac{4}{5}x^{5/4} - x + \frac{4}{3}x^{3/4} - 2x^{1/2} + 4x^{1/4} - 4\ln\left(1 + x^{1/4}\right) + C \] where \( C \) is the constant of integration. **Answer Summary:** After substitution, an antiderivative is (4⁄5)x^(5⁄4) – x + (4/3)x^(3/4) – 2 x^(1/2) + 4x^(1/4) minus four log terms. Thus, ∫√x/(1 + x^(1/4)) dx = (4/5)x^(5/4) − x + (4/3)x^(3/4) − 2x^(1/2) + 4x^(1/4) − 4 ln(1 + x^(1/4)) + C