Step 1: Identify the integral you need to solve, which is ∫_0^1 (x^4) dx.
Step 2: Use the power rule for integration. The power rule states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where n is a constant.
Step 3: In this case, n is 4. So, apply the power rule: ∫ x^4 dx = (x^(4+1))/(4+1) = (x^5)/5.
Step 4: Now, you need to evaluate this from 0 to 1. This means you will calculate (x^5)/5 at x = 1 and at x = 0.
Step 5: First, calculate at x = 1: (1^5)/5 = 1/5.
Step 6: Next, calculate at x = 0: (0^5)/5 = 0/5 = 0.
Step 7: Now, subtract the value at x = 0 from the value at x = 1: (1/5) - (0) = 1/5.
Step 8: Therefore, the value of the integral ∫_0^1 (x^4) dx is 1/5.
Definite Integral – The question tests the ability to evaluate a definite integral, which involves finding the area under the curve of the function x^4 from 0 to 1.
Power Rule for Integration – The solution requires the application of the power rule for integration, which states that ∫x^n dx = x^(n+1)/(n+1) + C.
