Integrate: `int e^x ((x^2+1)/(x+1)^2) dx.`

Answer

To solve the integral \( \int e^x \frac{x^2 + 1}{(x + 1)^2} \, dx \), we will use substitution and integration techniques. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = x + 1 \). Then, we have:
\[ x = t - 1 \quad \text{and} \quad dx = dt \]
Now, we can rewrite the integral in terms of \( t \):
\[ \int e^{t-1} \frac{(t-1)^2 + 1}{t^2} \, dt \]

### Step 2: Simplifying the Integrand
Next, we simplify the expression \( (t - 1)^2 + 1 \):
\[ (t - 1)^2 + 1 = t^2 - 2t + 1 + 1 = t^2 - 2t + 2 \]
Thus, the integral becomes:
\[ \int e^{t-1} \frac{t^2 - 2t + 2}{t^2} \, dt \]

### Step 3: Factor Out \( e^{-1} \)
We can factor out \( e^{-1} \):
\[ e^{-1} \int e^t \left( \frac{t^2 - 2t + 2}{t^2} \right) dt \]
This can be separated into:
\[ e^{-1} \int e^t \left( 1 - \frac{2}{t} + \frac{2}{t^2} \right) dt \]

### Step 4: Splitting the Integral
Now we can split the integral:
\[ e^{-1} \left( \int e^t \, dt - 2 \int \frac{e^t}{t} \, dt + 2 \int \frac{e^t}{t^2} \, dt \right) \]

### Step 5: Integrating Each Term
1. The first integral \( \int e^t \, dt = e^t + C_1 \).
2. The second integral \( \int \frac{e^t}{t} \, dt \) is a known integral, often denoted as \( \text{Ei}(t) \).
3. The third integral \( \int \frac{e^t}{t^2} \, dt \) can be solved using integration by parts.

### Step 6: Putting It All Together
Combining these results, we have:
\[ e^{-1} \left( e^t - 2 \text{Ei}(t) + 2 \left( -\frac{e^t}{t} + C_2 \right) \right) \]
Substituting back \( t = x + 1 \):
\[ e^{-1} \left( e^{x+1} - 2 \text{Ei}(x+1) - \frac{2e^{x+1}}{x+1} + C \right) \]

### Final Answer
Thus, the final answer is:
\[ \int e^x \frac{x^2 + 1}{(x + 1)^2} \, dx = e^x - \frac{2 e^x}{x + 1} + C \]