Evaluate the following integrals:
`int frac{x^(1backslash2)}{1+x^(3backslash4)}dx`

Answer

To evaluate the integral

\[ I = \int \frac{x^{1/2}}{1 + x^{3/4}} \, dx, \]

we will use substitution and integration techniques. Let's go through the steps:

### Step 1: Substitution
Let \( x = t^4 \). Then, we differentiate to find \( dx \):

\[ dx = 4t^3 \, dt. \]

### Step 2: Substitute in the integral
Now we substitute \( x \) and \( dx \) into the integral:

\[ I = \int \frac{(t^4)^{1/2}}{1 + (t^4)^{3/4}} \cdot 4t^3 \, dt. \]

This simplifies to:

\[ I = \int \frac{t^2}{1 + t^3} \cdot 4t^3 \, dt = 4 \int \frac{t^5}{1 + t^3} \, dt. \]

### Step 3: Polynomial Long Division
Now, we perform polynomial long division on \( \frac{t^5}{1 + t^3} \):

1. Divide \( t^5 \) by \( t^3 \) to get \( t^2 \).
2. Multiply \( t^2 \) by \( 1 + t^3 \) to get \( t^2 + t^5 \).
3. Subtract \( t^5 + t^2 \) from \( t^5 \) to get \( -t^2 \).

Thus, we can rewrite the integral as:

\[ I = 4 \int \left( t^2 - \frac{t^2}{1 + t^3} \right) dt. \]

### Step 4: Integrate
Now we integrate term by term:

1. The integral of \( t^2 \) is:

\[ \int t^2 \, dt = \frac{t^3}{3}. \]

2. For the second term, we need to integrate \( \frac{t^2}{1 + t^3} \). We can use substitution \( z = 1 + t^3 \), which gives \( dz = 3t^2 \, dt \) or \( dt = \frac{dz}{3t^2} \). Thus,

\[ \int \frac{t^2}{1 + t^3} \, dt = \frac{1}{3} \int \frac{1}{z} \, dz = \frac{1}{3} \ln |z| + C = \frac{1}{3} \ln |1 + t^3| + C. \]

### Step 5: Combine results
Putting it all together, we have:

\[ I = 4 \left( \frac{t^3}{3} - \frac{1}{3} \ln |1 + t^3| \right) + C. \]

This simplifies to:

\[ I = \frac{4}{3} t^3 - \frac{4}{3} \ln |1 + t^3| + C. \]

### Step 6: Substitute back for \( t \)
Recall that \( t = x^{1/4} \):

\[ I = \frac{4}{3} (x^{1/4})^3 - \frac{4}{3} \ln |1 + (x^{1/4})^3| + C. \]

This simplifies to:

\[ I = \frac{4}{3} x^{3/4} - \frac{4}{3} \ln |1 + x^{3/4}| + C. \]

### Final Answer
Thus, the final result for the integral is:

\[ \int \frac{x^{1/2}}{1 + x^{3/4}} \, dx = \frac{4}{3} x^{3/4} - \frac{4}{3} \ln |1 + x^{3/4}| + C. \]