Question

How do you find the integral

∫
1
x
2
⋅
√
x
2
−
9
d
x
?

Answer 1

To find the integral ∫ (1 / (x^2 * √(x^2 - 9))) dx, we can use a trigonometric substitution. Let x = 3sec(θ), solve for dx in terms of dθ, substitute values into the integral, integrate and then substitute back x.

Step-by-step explanation:

To find the integral ∫ (1 / (x^2 ⋅ √(x^2 - 9))) dx, we can use a trigonometric substitution. Let x = 3sec(θ), which implies dx = 3sec(θ)tan(θ) dθ. Substitute these values into the integral:

∫ (1 / (x^2 ⋅ √(x^2 - 9))) dx = ∫ (1 / (9sec^2(θ) ⋅ √(9sec^2(θ) - 9))) (3sec(θ)tan(θ) dθ)

= ∫ (tan(θ) / (3sec(θ) ⋅ 3tan(θ))) dθ

= ∫ (tan(θ) / (9sec(θ))) dθ

= ∫ (sin(θ) / (9cos(θ))) dθ

= 1/9 ∫ (sin(θ) / cos(θ)) dθ

= 1/9 ln|sec(θ) + tan(θ)| + C

Finally, substitute back x and simplifying:

∫ (1 / (x^2 ⋅ √(x^2 - 9))) dx = 1/9 ln|sec(θ) + tan(θ)| + C