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

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

Step 1: Identify the integral we need to solve: ∫_0^1 (x^4 + 2x^3 + x^2) dx.
Step 2: Break down the integral into separate parts: ∫_0^1 x^4 dx + ∫_0^1 2x^3 dx + ∫_0^1 x^2 dx.
Step 3: Calculate each integral separately.
Step 4: For ∫_0^1 x^4 dx, use the formula ∫ x^n dx = (x^(n+1))/(n+1). Here, n=4, so it becomes (x^5)/5.
Step 5: Evaluate (x^5)/5 from 0 to 1: (1^5)/5 - (0^5)/5 = 1/5 - 0 = 1/5.
Step 6: For ∫_0^1 2x^3 dx, first find ∫ x^3 dx = (x^4)/4, then multiply by 2: 2 * (x^4)/4 = (x^4)/2.
Step 7: Evaluate (x^4)/2 from 0 to 1: (1^4)/2 - (0^4)/2 = 1/2 - 0 = 1/2.
Step 8: For ∫_0^1 x^2 dx, use the formula: ∫ x^2 dx = (x^3)/3.
Step 9: Evaluate (x^3)/3 from 0 to 1: (1^3)/3 - (0^3)/3 = 1/3 - 0 = 1/3.
Step 10: Add all the results together: 1/5 + 1/2 + 1/3.
Step 11: Find a common denominator to add the fractions: The common denominator for 5, 2, and 3 is 30.
Step 12: Convert each fraction: 1/5 = 6/30, 1/2 = 15/30, 1/3 = 10/30.
Step 13: Add the converted fractions: 6/30 + 15/30 + 10/30 = 31/30.
Step 14: The final answer is 31/30.

Definite Integrals – The question tests the ability to evaluate a definite integral of a polynomial function over a specified interval.
Polynomial Integration – It assesses the understanding of integrating polynomial terms using the power rule.
Evaluation of Limits – The question requires evaluating the antiderivative at the upper and lower limits of integration.