Evaluate the integral ∫ (x^3 - x) dx. What is the result?

Evaluate the integral ∫ (x^3 - x) dx. What is the result?

• A. (1/4)x^4 - (1/2)x^2 + C
• B. (1/4)x^4 + (1/2)x^2 + C
• C. (1/3)x^4 - (1/2)x^2 + C
• D. (1/5)x^4 - (1/2)x^2 + C

Answer: (A)

To evaluate the integral ∫ (x^3 - x) dx, we apply the power rule of integration, which states that ∫ x^n dx = (1/(n+1)) x^(n+1) + C, where n is any real number other than -1. First, we separate the integral into two parts: ∫ (x^3 - x) dx = ∫ x^3 dx - ∫ x dx. Now we evaluate each integral separately: 1. For ∫ x^3 dx, we use the power rule: ∫ x^3 dx = (1/(3+1)) x^(3+1) = (1/4) x^4. 2. For ∫ x dx, we apply the power rule again: ∫ x dx = (1/(1+1)) x^(1+1) = (1/2) x^2. Combining these results gives: ∫ (x^3 - x) dx = (1/4)x^4 - (1/2)x^2 + C. The final result (1/4)x^4 - (1/2)x^2 + C matches the first choice

To evaluate the integral ∫ (x^3 - x) dx, we apply the power rule of integration, which states that ∫ x^n dx = (1/(n+1)) x^(n+1) + C, where n is any real number other than -1.

First, we separate the integral into two parts:

∫ (x^3 - x) dx = ∫ x^3 dx - ∫ x dx.

Now we evaluate each integral separately:

1. For ∫ x^3 dx, we use the power rule:

∫ x^3 dx = (1/(3+1)) x^(3+1) = (1/4) x^4.

2. For ∫ x dx, we apply the power rule again:

∫ x dx = (1/(1+1)) x^(1+1) = (1/2) x^2.

Combining these results gives:

∫ (x^3 - x) dx = (1/4)x^4 - (1/2)x^2 + C.

The final result (1/4)x^4 - (1/2)x^2 + C matches the first choice