Evaluate ∫₁³(x² + 3x + eˣ)dx

The given integral is: ∫₁³(x² + 3x + eˣ)dx

Using the property of definite integrals: ∫ₐᵇ(f(x) + g(x))dx = ∫ₐᵇf(x)dx + ∫ₐᵇg(x)dx, the above integral becomes:

∫₁³x²dx + ∫₁³3xdx + ∫₁³eˣdx
= x³/3 |₁³ + 3x²/2 |₁³ + eˣ |₁³

Substituting the limits and adding up to get the result, we get:
27/3 - 1/3 + 3/2(9 - 1) + (e³ - e)
= 26/3 + 12 + e³ - e
= 26/3 + 36/3 + e³ - e
= 62/3 + e³ - e