Step 1: Identify the integral to evaluate, which is ∫ from 0 to 1 of (x^4 + 2x^3) dx.
Step 2: Break the integral into two parts: ∫ from 0 to 1 of x^4 dx and ∫ from 0 to 1 of 2x^3 dx.
Step 3: Calculate the first integral, ∫ x^4 dx. The antiderivative of x^4 is (x^5)/5.
Step 4: Calculate the second integral, ∫ 2x^3 dx. The antiderivative of 2x^3 is (2x^4)/4, which simplifies to (x^4)/2.
Step 5: Combine the results from Step 3 and Step 4. The combined antiderivative is (x^5)/5 + (x^4)/2.
Step 6: Evaluate the combined antiderivative from 0 to 1. Substitute 1 into the expression: (1^5)/5 + (1^4)/2 = 1/5 + 1/2.
Step 7: Simplify the result from Step 6. Convert 1/2 to a fraction with a common denominator: 1/2 = 5/10. So, 1/5 + 1/2 = 1/5 + 5/10 = 2/10 + 5/10 = 7/10.
Step 8: The final answer is 7/10.
Definite Integral Evaluation – The process of calculating the area under a curve defined by a function over a specific interval.
Polynomial Integration – Applying the power rule for integration to polynomial functions.
