`int x ^(x ^(2)+1) (2ln x+1)dx `

Options

A. `x ^(2x ) +C`
B. `x ^(2) ln x+C`
C. `x ^((x ^(x)))+C`
D. `(x ^(3))^(x) +C`

Answer

To solve the integral \( \int x^{(x^2 + 1)} (2 \ln x + 1) \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = x^{x^2} \). Then we can express \( \ln t \) in terms of \( x \):
\[ \ln t = x^2 \ln x \]

### Step 2: Differentiate
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\ln t) = \frac{1}{t} \frac{dt}{dx} = 2x \ln x + x \]
This gives us:
\[ \frac{dt}{dx} = t (2x \ln x + x) \]

### Step 3: Rearranging
Now, we can rearrange this to express \( dx \):
\[ dx = \frac{dt}{t(2x \ln x + x)} \]

### Step 4: Substitute back into the integral
Substituting \( dx \) back into the integral, we have:
\[ \int x^{(x^2 + 1)} (2 \ln x + 1) \, dx = \int t \cdot \frac{dt}{t(2x \ln x + x)} = \int dt \]

### Step 5: Integrate
The integral of \( dt \) is simply:
\[ t + C \]

### Step 6: Substitute back for \( t \)
Now substitute back for \( t \):
\[ t = x^{x^2} \]
Thus, we have:
\[ \int x^{(x^2 + 1)} (2 \ln x + 1) \, dx = x^{x^2} + C \]

### Final Answer
The final answer is:
\[ \int x^{(x^2 + 1)} (2 \ln x + 1) \, dx = x^{x^2} + C \]