Final answer:
The integral ∫x³/(x² + 49) dx is solved using trigonometric substitution by setting x = 7tan(θ), which simplifies the integral into a form that can be evaluated using standard techniques.
Step-by-step explanation:
The student is asked to evaluate the integral ∫x³/(x² + 49) dx using trigonometric substitution. To do so, we recognize that the denominator suggests a substitution involving the tangent function, because 1 + tan²(θ) = sec²(θ), which is related to the Pythagorean identity. By setting x = 7tan(θ), we have dx = 7sec²(θ)dθ, and the integral becomes: ∫(7³tan³(θ))/(49tan²(θ) + 49) ⋅ 7sec²(θ)dθ = ∫(343tan³(θ)sec²(θ))/(49sec²(θ)) dθ. Simplifying gives us ∫7tan³(θ)dθ, which can be evaluated using standard trigonometric integral techniques.
After computing the integral in terms of θ, we must then revert back to the original variable x using the substitution we initially made. The final step is to add the constant of integration, C, according to the Fundamental Theorem of Calculus. Trigonometric substitution is a useful technique for evaluating integrals involving radicals or quadratic expressions in the integrand.