## The Integral of √(x² + a²)

The integral of √(x² + a²) is a classic example of an integral that can be solved using **trigonometric substitution**. This type of integral often arises in problems related to **geometry, physics, and engineering**.

Here's a step-by-step breakdown of how to solve this integral:

### 1. Trigonometric Substitution

The key to solving this integral is to recognize that the expression under the square root resembles the Pythagorean identity:

**sin²θ + cos²θ = 1**

To make use of this, we can substitute:

**x = a tan θ**

This substitution allows us to simplify the expression under the square root. Differentiating both sides, we get:

**dx = a sec²θ dθ**

### 2. Simplifying the Integral

Substituting x and dx in the original integral, we get:

```
∫√(x² + a²) dx = ∫√(a² tan²θ + a²) * a sec²θ dθ
```

Simplifying the expression under the square root:

```
∫√(a²(tan²θ + 1)) * a sec²θ dθ = ∫√(a² sec²θ) * a sec²θ dθ
```

Since sec²θ = 1 + tan²θ, we can simplify further:

```
∫ a secθ * a sec²θ dθ = a² ∫ sec³θ dθ
```

### 3. Solving the Integral of sec³θ

The integral of sec³θ requires a technique called **integration by parts**. Here's how it's done:

- Let u = secθ and dv = sec²θ dθ
- Then du = secθ tanθ dθ and v = tanθ

Applying the integration by parts formula:

```
∫ sec³θ dθ = secθ tanθ - ∫ tan²θ secθ dθ
```

Using the identity tan²θ = sec²θ - 1, we can rewrite the integral:

```
∫ sec³θ dθ = secθ tanθ - ∫ (sec²θ - 1) secθ dθ
```

Simplifying and solving for the integral of sec³θ:

```
∫ sec³θ dθ = secθ tanθ - ∫ sec³θ dθ + ∫ secθ dθ
2 ∫ sec³θ dθ = secθ tanθ + ln|secθ + tanθ|
∫ sec³θ dθ = (1/2)(secθ tanθ + ln|secθ + tanθ|)
```

### 4. Back Substitution

Now, we need to substitute back for θ in terms of x. From the initial substitution, we have:

**tan θ = x/a**

Using the Pythagorean identity, we can find:

**sec θ = √(1 + tan²θ) = √(1 + (x²/a²)) = √(x² + a²)/a**

Substituting these values back into the integral:

```
∫√(x² + a²) dx = a² * (1/2) * (√(x² + a²)/a * x/a + ln|√(x² + a²)/a + x/a|) + C
```

Simplifying, we get the final result:

```
∫√(x² + a²) dx = **(1/2) * (x√(x² + a²) + a² ln|x + √(x² + a²)|) + C**
```

Where C is the constant of integration.

### Conclusion

The integral of √(x² + a²) can be solved using trigonometric substitution and integration by parts. This technique is valuable for solving many other integrals involving square roots of quadratic expressions.