Final answer:
To evaluate the integral ∫ x³√(1+x²) dx, trigonometric substitution is used by setting x = tan(θ) and simplifying the integral into a new form that can be solved using trigonometric identities or other integration techniques.
Step-by-step explanation:
The student asked to evaluate the integral ∫ x³√(1+x²) dx using trigonometric substitution. This technique involves replacing the variable with a trigonometric expression to simplify the integral.
In this case, one common substitution for solving an integral of this form is to set x = tan(θ), which makes dx = sec²(θ) dθ. Then, √(1+x²) becomes √(1+tan²(θ)) = sec(θ). Applying this substitution, the integral transforms into ∫ tan³(θ)sec³(θ) sec²(θ) dθ, which can then be evaluated using further trigonometric identities or integration techniques such as integration by parts.