`int(x^(x))^(2)(1+logx)dx=`

Options

A. `((x^(x))^(2))/(2)+c`
B. `x^(x)+c`
C. `(x^(x))/(2)+c`
D. None of these

Answer

To solve the integral \( I = \int (x^x)^2 (1 + \log x) \, dx \), we will follow these steps:

### Step 1: Substitution
Let \( t = x^x \).

### Step 2: Differentiate \( t \)
To find \( \frac{dt}{dx} \), we take the natural logarithm of both sides:
\[ \log t = x \log x \]
Differentiating both sides using the product rule:
\[ \frac{1}{t} \frac{dt}{dx} = \log x + 1 \]
Thus, we have:
\[ \frac{dt}{dx} = t (\log x + 1) = x^x (1 + \log x) \]

### Step 3: Express \( dx \) in terms of \( dt \)
From the equation \( \frac{dt}{dx} = x^x (1 + \log x) \), we can express \( dx \):
\[ dx = \frac{dt}{x^x (1 + \log x)} \]

### Step 4: Substitute in the integral
Now we substitute \( t \) and \( dx \) into the integral:
\[ I = \int (x^x)^2 (1 + \log x) \, dx = \int t^2 \, \frac{dt}{x^x (1 + \log x)} \]
Since \( t = x^x \), we can replace \( x^x \) in the denominator:
\[ I = \int t^2 \, dt \]

### Step 5: Integrate
The integral \( \int t^2 \, dt \) is straightforward:
\[ I = \frac{t^3}{3} + C \]

### Step 6: Substitute back for \( t \)
Now, we substitute back \( t = x^x \):
\[ I = \frac{(x^x)^3}{3} + C = \frac{x^{3x}}{3} + C \]

### Final Result
Thus, the final answer is:
\[ I = \frac{x^{3x}}{3} + C \]

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