Mathematics

This is a quadratic form under a square root, which calls for a trigonometric substitution to simplify the radical. Recognize that $16 + 9x^2$ has the structure $a^2 + (bx)^2$, so substituting $x = \frac{a}{b}\tan\theta$ will convert the sum into a perfect square under the radical. Once the square root simplifies, you'll integrate a trigonometric expression, then convert back to $x$ using a reference triangle.

This is a quadratic form under a square root, which calls for a trigonometric substitution to simplify the radical. Recognize that $16 + 9x^2$ has the structure $a^2 + (bx)^2$, so substituting $x = \frac{a}{b}\tan\theta$ will convert the sum into a perfect square under the radical. Once the square root simplifies, you'll integrate a trigonometric expression, then convert back to $x$ using a reference triangle.