Step-by-Step: The Power Rule for Integration

1. Add 1 to the exponent.
2. Divide the coefficient by the new exponent.
3. Add the constant of integration, + C.

Worked Example

Integrate f(x) = 4x³.

Add 1 to exponent: 3+1=4.

Divide coefficient by new exponent: 4/4 = 1.

∫4x³ dx = x⁴ + C

For a definite integral from 0 to 2: [x⁴] evaluated at 2 minus at 0 = 16 − 0 = 16.

Frequently Asked Questions

What happens when the exponent is −1?

The power rule breaks down (division by zero), so ∫(1/x)dx uses a special case instead: the natural logarithm, ln|x| + C.

How do definite integrals differ in the calculation?

After finding the antiderivative, evaluate it at the upper limit, then subtract its value at the lower limit — no "+ C" is needed since the constant cancels out in the subtraction.