Evaluate `inte^(3logx)(x^(4)+1)^(-1)dx`

Answer

To evaluate the integral

\[ \int e^{3 \log x} (x^4 + 1)^{-1} \, dx, \]

we can follow these steps:

### Step 1: Simplify the Expression
We start by simplifying \( e^{3 \log x} \). Using the property of logarithms, we know that \( e^{\log a} = a \). Therefore, we can rewrite \( e^{3 \log x} \) as:

\[ e^{3 \log x} = (e^{\log x})^3 = x^3. \]

So, the integral becomes:

\[ \int \frac{x^3}{x^4 + 1} \, dx. \]

### Step 2: Substitution
Next, we can use substitution to simplify the integral further. Let:

\[ t = x^4 + 1. \]

Now, we differentiate both sides with respect to \( x \):

\[ \frac{dt}{dx} = 4x^3 \implies dt = 4x^3 \, dx \implies dx = \frac{dt}{4x^3}. \]

### Step 3: Substitute in the Integral
Now, we substitute \( t \) and \( dx \) into the integral:

\[ \int \frac{x^3}{t} \cdot \frac{dt}{4x^3} = \int \frac{1}{4t} \, dt. \]

### Step 4: Integrate
Now we can integrate:

\[ \int \frac{1}{4t} \, dt = \frac{1}{4} \ln |t| + C, \]

where \( C \) is the constant of integration.

### Step 5: Substitute Back
Finally, we substitute back \( t = x^4 + 1 \):

\[ \frac{1}{4} \ln |x^4 + 1| + C. \]

### Final Answer
Thus, the evaluated integral is:

\[ \int e^{3 \log x} (x^4 + 1)^{-1} \, dx = \frac{1}{4} \ln |x^4 + 1| + C. \]