Evaluate ∫ from 1 to 2 of (x^4 - 4x^3 + 6x^2 - 4x + 1) dx.
Practice Questions
Q1
Evaluate ∫ from 1 to 2 of (x^4 - 4x^3 + 6x^2 - 4x + 1) dx.
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Questions & Step-by-Step Solutions
Evaluate ∫ from 1 to 2 of (x^4 - 4x^3 + 6x^2 - 4x + 1) dx.
Step 1: Identify the function to integrate, which is f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1.
Step 2: Find the antiderivative (indefinite integral) of f(x). This means we need to integrate each term separately.
Step 3: Integrate each term: The integral of x^4 is (x^5)/5, the integral of -4x^3 is -x^4, the integral of 6x^2 is 2x^3, the integral of -4x is -2x^2, and the integral of 1 is x.
Step 4: Combine the results of the integrals: The antiderivative is (x^5)/5 - x^4 + 2x^3 - 2x^2 + x.
Step 5: Evaluate the definite integral from 1 to 2. This means we will calculate the antiderivative at 2 and at 1, and then subtract the two results.
Step 6: Calculate the antiderivative at x = 2: (2^5)/5 - (2^4) + 2(2^3) - 2(2^2) + 2.
Step 7: Calculate the antiderivative at x = 1: (1^5)/5 - (1^4) + 2(1^3) - 2(1^2) + 1.
Step 8: Subtract the result from Step 7 from the result from Step 6 to find the value of the definite integral.
Step 9: The final result is 0.
Definite Integral Evaluation – The question tests the ability to evaluate a definite integral of a polynomial function.
Polynomial Integration – It assesses knowledge of integrating polynomial functions and applying the Fundamental Theorem of Calculus.
