Question

Evaluate the integral ∫dx/(x² +4x+13)¹/² by completing the square and using trigonometric substitution.

Answer 1

To evaluate the given integral, complete the square and use trigonometric substitution.

Step-by-step explanation:

To evaluate the integral ∫dx/(x² +4x+13)¹/² using completing the square and trigonometric substitution, follow these steps:

1. Complete the square by rewriting the expression as ∫dx/((x+2)² +9).
2. Make a trigonometric substitution such as x+2 = 3tan(θ), which leads to dx = 3sec²(θ)dθ.
3. Substituting these values in the integral gives ∫3sec²(θ)dθ/((3tan(θ))²+9).
4. Simplify the expression using trigonometric identities, integrate with respect to θ, and change the variable back to x to obtain the final answer.