Final answer:
To find the indefinite integral of the function ∫e^x (1−e^2x) dx, we can use the method of integration by substitution. Let's make the substitution u = 1 - e^2x. The indefinite integral is -u^2/4 + C, where u = 1 - e^2x and C is the constant of integration.
Step-by-step explanation:
To find the indefinite integral of the function ∫e^x (1−e^2x) dx, we can use the method of integration by substitution. Let's make the substitution u = 1 - e^2x. The differential of u is given by du = -2e^2x dx. Rearranging this equation, we find that dx = -du/(2e^2x).
Substituting the values of u and dx into the original integral, we get:
∫e^x (1−e^2x) dx = ∫e^x (u) (-du/(2e^2x)) = -∫(u/2) du = -u^2/4 + C
Therefore, the indefinite integral of the given function is -u^2/4 + C, where u = 1 - e^2x and C is the constant of integration.