Evaluate: ∫ √(x² − a²) d x

To evaluate the integral ∫ √(x² − a²) dx, we can use a trigonometric substitution. This method simplifies the expression under the square root.

Step-by-Step Solution

1. Trigonometric Substitution

Let’s set:

• x = a sec(θ)
• dx = a sec(θ) tan(θ) dθ

With this substitution, we can rewrite the integral:

√(x² − a²) becomes √(a² sec²(θ) − a²) = √(a² (sec²(θ) − 1)) = a tan(θ).

2. Rewrite the Integral

The integral now transforms to:

∫ a tan(θ) (a sec(θ) tan(θ) dθ) = a² ∫ tan²(θ) sec(θ) dθ.

3. Use the Identity

Recall that tan²(θ) = sec²(θ) - 1. Thus, we can split the integral:

∫ tan²(θ) sec(θ) dθ = ∫ (sec²(θ) - 1) sec(θ) dθ = ∫ sec³(θ) dθ - ∫ sec(θ) dθ.

4. Integrate

The integral of sec(θ) is ln |sec(θ) + tan(θ)|, and the integral of sec³(θ) can be solved using integration by parts:

• ∫ sec³(θ) dθ = (1/2)(sec(θ) tan(θ) + ln |sec(θ) + tan(θ)|).

5. Combine Results

Putting it all together, we have:

∫ √(x² − a²) dx = (1/2)(sec(θ) tan(θ) + ln |sec(θ) + tan(θ)|) - ln |sec(θ) + tan(θ)| + C.

Final Result

Now, substitute back θ using x = a sec(θ):

• sec(θ) = x/a
• tan(θ) = √(x² − a²)/a.

The final answer is:

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