Step 1: Identify the function to integrate, which is x^3 + 2x.
Step 2: Break the integral into two parts: ∫(x^3 + 2x)dx = ∫x^3dx + ∫2xdx.
Step 3: Integrate the first part, ∫x^3dx. Use the power rule: add 1 to the exponent (3 + 1 = 4) and divide by the new exponent. This gives (1/4)x^4.
Step 4: Integrate the second part, ∫2xdx. Again, use the power rule: add 1 to the exponent (1 + 1 = 2) and divide by the new exponent. This gives (2/2)x^2, which simplifies to x^2.
Step 5: Combine the results from Step 3 and Step 4. You get (1/4)x^4 + x^2.
Step 6: Don't forget to add the constant of integration, C. So the final answer is (1/4)x^4 + x^2 + C.
