Can you demonstrate how to find the indefinite integral of the function g(x) = e^(2x)?
user
Sure, I'd be happy to help with that. The indefinite integral of a function is an antiderivative of that function. In other words, it's a function whose first derivative is equal to the original function.

For g(x) = e^(2x), the indefinite integral can be found using the substitution u = 2x. Using the chain rule, if y is a function of u, we have:

    dy/dx = (dy/du) * (du/dx)

    In this case, we have y = e^u, so dy/du = e^u, and du/dx = 2, so:

    dy/dx = e^(2x) * 2 = 2e^(2x)

    This shows that the indefinite integral of e^(2x) is f(x) = (1/2)e^(2x) + C, where C is a constant of integration.

Therefore, the indefinite integral of g(x) = e^(2x) is F(x) = (1/2)e^(2x) + C.
assistant
Can you demonstrate how to find the indefinite integral of the function g(x) = e^(2x)?
user
Sure, I'd be happy to help with that. The indefinite integral of a function involves finding a function whose derivative is equal to the original function. In this case, we're looking for the indefinite integral of g(x) = e^(2x).

The integral of e^(ax) is given by the formula 1/a * e^(ax) + C, where a is a constant and C is the constant of integration.

So, in this case, a is 2. Therefore, the indefinite integral of e^(2x) is:

∫e^(2x) dx = (1/2) * e^(2x) + C

Where C is the constant of integration.
assistant