Find an antiderivative using substitution or integration by parts, then compute the area directly with very thin rectangles (a Riemann sum) and verify the two values really match.
An expression like ∫2x(x²+3)⁴dx can feel overwhelming at first. That's exactly when substitution comes in handy. If an expression contains "a function paired with its own derivative," you can swap that entire function out for a new variable, u, turning it into a much simpler expression. Set u=x²+3, and du=2x dx, and the original expression turns into the simple power integral ∫u⁴du.
When two different kinds of functions are multiplied together (for example, x·eˣ), you use integration by parts: the formula ∫u dv = uv − ∫v du. You can check directly whether the answers from these two methods are really correct by taking a Riemann sum — approximating the area by adding up the areas of very thin rectangles across the interval. Check for yourself that the two values come out almost identical.