Integration of certain irrational functions with the aid of trigonometric or hyperbolic substitutions
`I=int(sqrt(x^(2)+1))/(x^(2))dx.`

Answer

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

### Step 1: Trigonometric Substitution
We start by substituting \( x = \tan(\theta) \). This gives us:
\[ dx = \sec^2(\theta) \, d\theta \]
Also, we have:
\[ \sqrt{x^2 + 1} = \sqrt{\tan^2(\theta) + 1} = \sqrt{\sec^2(\theta)} = \sec(\theta) \]

### Step 2: Rewrite the Integral
Substituting these into the integral, we get:
\[ I = \int \frac{\sec(\theta)}{\tan^2(\theta)} \sec^2(\theta) \, d\theta \]
This simplifies to:
\[ I = \int \frac{\sec^3(\theta)}{\tan^2(\theta)} \, d\theta \]

### Step 3: Express in Terms of Sine and Cosine
Recall that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Therefore, we can rewrite the integral as:
\[ I = \int \frac{1}{\cos^3(\theta)} \cdot \frac{\cos^2(\theta)}{\sin^2(\theta)} \, d\theta = \int \frac{1}{\sin^2(\theta) \cos(\theta)} \, d\theta \]

### Step 4: Use the Identity for Integration
Now, we can use the identity \( \csc^2(\theta) = \frac{1}{\sin^2(\theta)} \):
\[ I = \int \csc^2(\theta) \sec(\theta) \, d\theta \]

### Step 5: Integrate
The integral of \( \csc^2(\theta) \) is known:
\[ \int \csc^2(\theta) \, d\theta = -\cot(\theta) \]
Thus, we can write:
\[ I = -\cot(\theta) \sec(\theta) + C \]

### Step 6: Back Substitute
Now we need to convert back to \( x \). Recall that:
\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{x} \]
and
\[ \sec(\theta) = \sqrt{x^2 + 1} \]
Thus:
\[ I = -\frac{\sqrt{x^2 + 1}}{x} + C \]

### Final Answer
The final result for the integral is:
\[ I = -\frac{\sqrt{x^2 + 1}}{x} + C \]