Question

If f'(x) = 4xe^(x²) and f(0) = 8, find f(x).

Answer 1

To find the function f(x), integrate f'(x) = 4xe^(x²) with respect to x and use the initial condition f(0) = 8. The function f(x) = 2e^(x²) + 6.

Step-by-step explanation:

To find the function f(x), we need to integrate f'(x) with respect to x. Since f'(x) = 4xe^(x²), we integrate this expression to get f(x). Let's start by finding the antiderivative of 4xe^(x²):

∫4xe^(x²) dx = ∫2(2x)e^(x²) dx = 2 ∫2xe^(x²) dx

We can use the substitution method with u = x² and du = 2xdx. This allows us to rewrite the integral as:

2 ∫e^u du

Integrating e^u gives us:

2e^u + C

Substituting back the value of u, we get:

2e^(x²) + C

Now we need to find the specific value of C by using the given initial condition f(0) = 8:

f(0) = 2e^(0²) + C = 2e^0 + C = 2 + C = 8

Therefore, C = 6.

So the function f(x) is:

f(x) = 2e^(x²) + 6