The integral of x^3 is (1/4)x^4 and the integral of -4x is -2x^2. Therefore, ∫ (x^3 - 4x) dx = (1/4)x^4 - 2x^2 + C.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate the integral ∫ (x^3 - 4x) dx.
Solution: The integral of x^3 is (1/4)x^4 and the integral of -4x is -2x^2. Therefore, ∫ (x^3 - 4x) dx = (1/4)x^4 - 2x^2 + C.
Steps: 6
Step 1: Identify the integral you need to calculate: ∫ (x^3 - 4x) dx.
Step 2: Break the integral into two parts: ∫ x^3 dx and ∫ -4x dx.
Step 3: Calculate the integral of x^3. The formula for the integral of x^n is (1/(n+1))x^(n+1). Here, n is 3, so: ∫ x^3 dx = (1/(3+1))x^(3+1) = (1/4)x^4.
Step 4: Calculate the integral of -4x. Use the same formula: ∫ -4x dx = -4 * (1/(1+1))x^(1+1) = -4 * (1/2)x^2 = -2x^2.
Step 5: Combine the results from Step 3 and Step 4: ∫ (x^3 - 4x) dx = (1/4)x^4 - 2x^2.
Step 6: Don't forget to add the constant of integration, C, to the final answer: ∫ (x^3 - 4x) dx = (1/4)x^4 - 2x^2 + C.
