Find the antiderivative of the exponential function e^x*√1+e^x.

Question

Find the antiderivative of the exponential function e^x*√1+e^x.

Answer 1

To find the antiderivative of the function e^x * √(1+e^x), we will use a substitution method.

Let's start by making the substitution:

u = 1 + e^x

du/dx = e^x

Now, we can rewrite our original function in terms of u:

∫ e^x * √(1+e^x) dx = ∫ √u * du

Now we need to find the antiderivative of √u with respect to u:

∫ √u * du = ∫ u^(1/2) du

Using the power rule for integration, we get: (2/3) * u^(3/2) + C

Now we need to substitute back for u in terms of x:

(2/3) * (1+e^x)^(3/2) + C

So the antiderivative of e^x * √(1+e^x) is: (2/3) * (1+e^x)^(3/2) + C