Final answer:
To solve the integral ∫(98x)/(x²+4)√dx using the substitution x=2tan(θ), we change the integral into a trigonometric form that can be integrated using standard methods.
Step-by-step explanation:
To evaluate the indefinite integral ∫98xx2x2+4√ using the substitution x=2tan(θ), we first notice that the integrand simplifies based on trigonometric identities. When x=2tan(θ), the integral's denominator becomes x²+4 = 4tan²(θ)+4, which simplifies to 4sec²(θ). Furthermore, the derivative of 2tan(θ) with respect to θ is 2sec²(θ), so dx can be replaced with 2sec²(θ) dθ in the integral. Thus, making the substitution leads to a new integral that we can evaluate using standard trigonometric integrals.