Evaluate the integral: ∫ (2x^3 - 3x^2 + 4) dx

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Question: Evaluate the integral: ∫ (2x^3 - 3x^2 + 4) dx

Options:

Correct Answer: (1/2)x^4 - x^3 + 4x + C

Solution:

Integrating term by term gives (1/4)x^4 - (1/3)x^3 + 4x + C.

Evaluate the integral: ∫ (2x^3 - 3x^2 + 4) dx

Evaluate the integral: ∫ (2x^3 - 3x^2 + 4) dx

Correct Answer: (1/4)x^4 - (1/3)x^3 + 4x + C

Step 1: Identify the function to integrate, which is 2x^3 - 3x^2 + 4.
Step 2: Break down the integral into separate parts: ∫(2x^3) dx, ∫(-3x^2) dx, and ∫(4) dx.
Step 3: Integrate each part one by one.
Step 4: For ∫(2x^3) dx, use the power rule: increase the exponent by 1 (3+1=4) and divide by the new exponent. This gives (2/4)x^4 = (1/2)x^4.
Step 5: For ∫(-3x^2) dx, again use the power rule: increase the exponent by 1 (2+1=3) and divide by the new exponent. This gives (-3/3)x^3 = -x^3.
Step 6: For ∫(4) dx, the integral of a constant is the constant multiplied by x. So, this gives 4x.
Step 7: Combine all the integrated parts: (1/2)x^4 - x^3 + 4x.
Step 8: Don't forget to add the constant of integration, C, at the end. So the final answer is (1/2)x^4 - x^3 + 4x + C.

Integration – The process of finding the integral of a function, which involves applying the power rule to each term.
Power Rule – A rule used in integration that states ∫x^n dx = (1/(n+1))x^(n+1) + C for n ≠ -1.