What is `int_0^1 x(1-x)^9 dx` equal to

Options

A. `1//110`
B. `1//132`
C. `1//148`
D. `1//240`

Answer

To solve the integral \( \int_0^1 x(1-x)^9 \, dx \), we can use integration by parts. Let's go through the steps systematically.

### Step 1: Identify \( u \) and \( dv \)
We will use the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Let:
- \( u = x \)
- \( dv = (1-x)^9 \, dx \)

### Step 2: Differentiate \( u \) and integrate \( dv \)
Now, we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = dx \]
- Integrate \( dv \):
\[ v = \int (1-x)^9 \, dx \]
Using the power rule for integration:
\[ v = -\frac{(1-x)^{10}}{10} \]

### Step 3: Apply the integration by parts formula
Now we can apply the integration by parts formula:
\[ \int_0^1 x(1-x)^9 \, dx = \left[ x \cdot \left(-\frac{(1-x)^{10}}{10}\right) \right]_0^1 - \int_0^1 \left(-\frac{(1-x)^{10}}{10}\right) \, dx \]
This simplifies to:
\[ = -\frac{1}{10} \left[ x(1-x)^{10} \right]_0^1 + \frac{1}{10} \int_0^1 (1-x)^{10} \, dx \]

### Step 4: Evaluate the boundary term
Now we evaluate the boundary term:
\[ \left[ x(1-x)^{10} \right]_0^1 = 1 \cdot 0^{10} - 0 \cdot 1^{10} = 0 \]
Thus, the first term is zero.

### Step 5: Evaluate the remaining integral
Now we need to evaluate:
\[ \frac{1}{10} \int_0^1 (1-x)^{10} \, dx \]
Using the power rule for integration:
\[ \int (1-x)^{10} \, dx = -\frac{(1-x)^{11}}{11} \]
Evaluating from 0 to 1:
\[ = -\frac{(1-x)^{11}}{11} \bigg|_0^1 = -\left(0 - \frac{1^{11}}{11}\right) = \frac{1}{11} \]

### Step 6: Combine results
Now substituting back:
\[ \int_0^1 x(1-x)^9 \, dx = 0 + \frac{1}{10} \cdot \frac{1}{11} = \frac{1}{110} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_0^1 x(1-x)^9 \, dx = \frac{1}{110} \]

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