Integration

Power Rule for Integration

Find ∫x³ dx by applying the power rule for integration to get x⁴ / 4 + C.

antiderivative-power-rule The power rule for integration states that ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1. The antiderivative F satisfies d/dx[F(x)] = f(x).

Example

Increase the exponent, divide by the new exponent, and include plus C.

highlighted = computed this step

Step 1 — Set up

Set up the indefinite integral.

∫x3 dx\int x^{3} \,dx

Step 2 — Power rule

Use the power rule and include plus C.

∫xn dx=xn+1n+1+C\int x^n\,dx=\frac{x^{n+ 1 }}{n+ 1 }+C

Step 3 — Add one to exponent

Add 1 to the exponent to get 4 over 4.

x44+C\frac{x^{ \hl{4} }}{ \hl{4} }+C

Step 4 — Result

State the antiderivative with plus C.

x44+C\hlmath{\frac{x^{4}}{4}} +C