What's the integral of e^(2x)?

The integral of e^(2x) is (1/2)e^(2x) + C, where C is the constant of integration.

To find the integral of e^(2x), we can use the power rule of integration, which states that the integral of x^n is (1/(n+1))x^(n+1) + C. However, since e^(2x) is not in the form x^n, we need to use a different method.

We can use substitution to solve the integral of e^(2x). Let u = 2x, then du/dx = 2 and dx = (1/2)du. Substituting these into the integral, we get:

∫e^(2x) dx = ∫e^u (1/2)du
= (1/2) ∫e^u du
= (1/2) e^u + C
= (1/2) e^(2x) + C

A-Level Maths Tutor Summary: In simple terms, to find the integral of e^2x, we substitute u = 2x. This changes the integral to 1/2∫e^udu, which is straightforward to solve. The answer is 1/2​e^2x+C, where C is a constant. This method shows how substitution can simplify and help solve integrals that don't fit the usual patterns.