The integral ∫(2x^3 - 4x + 1)dx = (1/2)x^4 - 2x^2 + x + C.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the integral of f(x) = 2x^3 - 4x + 1.
Solution: The integral ∫(2x^3 - 4x + 1)dx = (1/2)x^4 - 2x^2 + x + C.
Steps: 9
Step 1: Identify the function you want to integrate, which is f(x) = 2x^3 - 4x + 1.
Step 2: Write down the integral you need to solve: ∫(2x^3 - 4x + 1)dx.
Step 3: Integrate each term of the function separately.
Step 4: For the first term, 2x^3, use the power rule: increase the exponent by 1 (3 + 1 = 4) and divide by the new exponent (4). This gives (2/4)x^4 = (1/2)x^4.
Step 5: For the second term, -4x, again use the power rule: increase the exponent by 1 (1 + 1 = 2) and divide by the new exponent (2). This gives (-4/2)x^2 = -2x^2.
Step 6: For the last term, +1, the integral of a constant is just the constant times x. So, the integral of +1 is +x.
Step 7: Combine all the integrated terms: (1/2)x^4 - 2x^2 + x.
Step 8: Don't forget to add the constant of integration, C, at the end.
Step 9: Write the final answer: ∫(2x^3 - 4x + 1)dx = (1/2)x^4 - 2x^2 + x + C.
