`int_0^1 (x)/((1 - x)^(3//4)) dx` =

Options

A. `12//5`
B. `16//5`
C. `-16//5`
D. none of these

Answer

To solve the integral \( \int_0^1 \frac{x}{(1 - x)^{3/4}} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We can rewrite the integral as:
\[ \int_0^1 \frac{x}{(1 - x)^{3/4}} \, dx = \int_0^1 x (1 - x)^{-3/4} \, dx \]

### Step 2: Use Integration by Parts
We can use integration by parts, where we let:
- \( u = x \) (thus \( du = dx \))
- \( dv = (1 - x)^{-3/4} \, dx \)

Now we need to find \( v \) by integrating \( dv \):
\[ v = \int (1 - x)^{-3/4} \, dx \]

### Step 3: Integrate \( dv \)
To integrate \( (1 - x)^{-3/4} \), we can use the power rule:
\[ \int (1 - x)^{-3/4} \, dx = \frac{(1 - x)^{1/4}}{1/4} = 4(1 - x)^{1/4} + C \]

### Step 4: Apply Integration by Parts Formula
Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ \int_0^1 x (1 - x)^{-3/4} \, dx = \left[ x \cdot 4(1 - x)^{1/4} \right]_0^1 - \int_0^1 4(1 - x)^{1/4} \, dx \]

### Step 5: Evaluate the Boundary Terms
Now we evaluate the boundary term:
\[ \left[ x \cdot 4(1 - x)^{1/4} \right]_0^1 = 4 \cdot 1 \cdot (1 - 1)^{1/4} - 4 \cdot 0 \cdot (1 - 0)^{1/4} = 0 - 0 = 0 \]

### Step 6: Evaluate the Remaining Integral
Now we need to evaluate:
\[ - \int_0^1 4(1 - x)^{1/4} \, dx \]
Using the power rule again:
\[ \int (1 - x)^{1/4} \, dx = \frac{(1 - x)^{5/4}}{5/4} = \frac{4}{5}(1 - x)^{5/4} + C \]
Thus:
\[ \int_0^1 4(1 - x)^{1/4} \, dx = 4 \cdot \left[ \frac{4}{5}(1 - x)^{5/4} \right]_0^1 = 4 \cdot \left( 0 - \frac{4}{5} \right) = -\frac{16}{5} \]

### Step 7: Combine Results
Putting it all together:
\[ \int_0^1 x (1 - x)^{-3/4} \, dx = 0 - \left( -\frac{16}{5} \right) = \frac{16}{5} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_0^1 \frac{x}{(1 - x)^{3/4}} \, dx = \frac{16}{5} \]