Calculate ∫ x3 dx. A. 3x2 + C B. x4/4 + C C. x2 D. x4 + C

In order to find the integral, we first take the anti-derivative. Imagine that x³ is the derivative of an original function. What would that function be? Based on our rules for taking the derivative, it would have an exponent that is 1 greater the 3, so 4. It also would have had a coefficient that, when multiplied by 4, made the derivative have a coefficient of 1, since the derivative is 1 * x³

Therefore:

anti-derivative (x³) = (1/4) x⁴

To confirm, take the derivative of (1/4) x⁴ and make sure it equals x³ (it does).

Wait a minute, how do we know that there wasn't some constant in the original function? We can't tell because because the derivative of a constant is zero, so we can't determine it based on the derivative that we have. For this reason, we add our constant of integration, C.

Answer:

B. (1/4) x⁴ + C

Check our work! Take the derivative our answer: Do you get x³?