The integral ∫(2x^3 - 4x)dx = (1/2)x^4 - 2x^2 + C.
Questions & Step-by-step Solutions
1 item
Q
Q: Evaluate the integral ∫(2x^3 - 4x)dx.
Solution: The integral ∫(2x^3 - 4x)dx = (1/2)x^4 - 2x^2 + C.
Steps: 7
Step 1: Identify the integral you need to evaluate: ∫(2x^3 - 4x)dx.
Step 2: Break the integral into two parts: ∫(2x^3)dx and ∫(-4x)dx.
Step 3: For the first part, ∫(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 4: For the second part, ∫(-4x)dx, again use the power rule: increase the exponent by 1 (1 + 1 = 2) and divide by the new exponent. This gives (-4/2)x^2 = -2x^2.
Step 5: Combine the results from Step 3 and Step 4: (1/2)x^4 - 2x^2.
Step 6: Don't forget to add the constant of integration, C, to the final answer.
Step 7: Write the final answer: ∫(2x^3 - 4x)dx = (1/2)x^4 - 2x^2 + C.
