Final answer:
The function f(x) defined as f(x) = ∫₀ˣ (t³ + 6t² + 6)dt is given by f(x) = (1/4)x⁴ + 2x³ + 6x.
Step-by-step explanation:
To find the function f(x) defined as f(x) = ∫₀ˣ (t³ + 6t² + 6)dt, we can evaluate the definite integral step by step.
- 1. Let's start by finding the antiderivative of the integrand, which is the function inside the integral symbol. The antiderivative of t³ is (1/4)t⁴, the antiderivative of 6t² is 2t³, and the antiderivative of 6 is 6t.
- 2. The indefinite integral of (t³ + 6t² + 6) is (1/4)t⁴ + 2t³ + 6t + C, where C is the constant of integration.
- 3. Now, we evaluate the definite integral from 0 to x by substituting the upper limit (x) into the antiderivative and subtracting the result when the lower limit (0) is substituted.
- 4. Plugging in x into the antiderivative gives us (1/4)x⁴ + 2x³ + 6x + C.
- 5. Next, we substitute 0 into the antiderivative to get (1/4)(0)⁴ + 2(0)³ + 6(0) + C, which simplifies to C.
- 6. Subtracting the result from step 5 from the result from step 4, we get f(x) = (1/4)x⁴ + 2x³ + 6x + C - C.
- 7. Simplifying further, f(x) = (1/4)x⁴ + 2x³ + 6x.
Therefore, the function f(x) defined as f(x) = ∫₀ˣ (t³ + 6t² + 6)dt is given by f(x) = (1/4)x⁴ + 2x³ + 6x.
Your question is incomplete, but most probably the full question was:
Find the function f(x) defined as f(x) = ∫₀ˣ (t³ + 6t² + 6)dt