Find the value of ∫_0^1 (x^4 + 2x^3) dx.

Find the value of ∫_0^1 (x^4 + 2x^3) dx.

Step 1: Identify the integral we need to solve: ∫_0^1 (x^4 + 2x^3) dx.
Step 2: Break the integral into two parts: ∫_0^1 x^4 dx + ∫_0^1 2x^3 dx.
Step 3: Find the antiderivative of x^4, which is (x^5)/5.
Step 4: Find the antiderivative of 2x^3, which is (2/4)x^4 or (1/2)x^4.
Step 5: Combine the antiderivatives: (x^5)/5 + (1/2)x^4.
Step 6: Evaluate the combined antiderivative from 0 to 1: [(1^5)/5 + (1/2)(1^4)] - [(0^5)/5 + (1/2)(0^4)].
Step 7: Calculate the values: (1/5 + 1/2) - (0 + 0).
Step 8: Convert 1/2 to a fraction with a common denominator: 1/2 = 5/10.
Step 9: Add the fractions: 1/5 + 5/10 = 2/10 + 5/10 = 7/10.

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