Find the value of ∫_0^1 (4x^3) dx.

Find the value of ∫_0^1 (4x^3) dx.

Step 1: Identify the integral you need to solve, which is ∫_0^1 (4x^3) 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: Apply the power rule to 4x^3. First, integrate x^3: ∫ x^3 dx = (x^(3+1))/(3+1) = (x^4)/4.
Step 4: Multiply by the constant 4: ∫ 4x^3 dx = 4 * (x^4)/4 = x^4.
Step 5: Now, evaluate the definite integral from 0 to 1. This means you will calculate x^4 at the upper limit (1) and the lower limit (0).
Step 6: Calculate at the upper limit: (1)^4 = 1.
Step 7: Calculate at the lower limit: (0)^4 = 0.
Step 8: Subtract the lower limit result from the upper limit result: 1 - 0 = 1.
Step 9: Therefore, the value of the integral ∫_0^1 (4x^3) dx is 1.

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