Evaluate the integral: ∫ e√3x dx
Solution: According to the given statement, we substitute u = √3x. If the expression ax + b occurs, we use the rationalizing substitution u = ax + b. More generally, this sometimes works for ng(x). Then, 3x = u^2, so dx = (1/2√3) du, and e√3x dx = (2/3)ue^u du.
The integrand is now a product of u and the transcendental function e^u, so it can be integrated by parts. Choosing u' = u and dv' = e^u du, we can proceed with the integration.