Question

For each integral, indicate what trig function would be used to solve using trigonometric substitution. (No work to show.) a. ∫ x² / x³+4 dx = tangent b. ∫ x² / 16 - 9x² dx = secant c. ∫ x² / 5x² - 3 dx = sine

Answer 1

For the given integrals, trigonometric functions such as tangent, secant, and sine are used respectively for making the trigonometric substitution to simplify and solve the integrals.

Step-by-step explanation:

Trigonometric Substitution for Integrals

For the integral ∫ x² / (x³+4) dx, one can use the tangent function for trigonometric substitution. In cases where the denominator is in the form of a² + x², it is common to let x = a tan(θ), simplifying the integration process.

For the integral ∫ x² / (16 - 9x²) dx, the secant function should be appointed. Here, where the denominator has the form of a² - x², the substitution x = a sec(θ) is appropriate.

Lastly, for ∫ x² / (5x² - 3) dx, one might leverage the sine function. When the expression includes x² - a², a useful substitution is x = a sin(θ)