# Indefinite Integration of Polynomials

How to find the indefinite integration of a polynomial: formulas, examples, and their solutions.

## Formula: Constant Multiple Rule in Integration

The coefficient *k* is not affected

by the integration.

(Just like the coefficient in differentiation.)

So ∫ *kf*(*x*) *dx* = *k* ∫ *f*(*x*) *dx*.

This formula is needed

to find the integrals of polynomials.

## Formula: Sum Rule (and Subtraction Rule) in Integration

The plus minus signs (±) are not affected

by the integration.

(Just like the plus minus signs in differentiation.)

So ∫ [*f*(*x*) ± *g*(*x*)] *dx* = ∫ *f*(*x*) *dx* ± ∫ *g*(*x*) *dx*.

This formula is also needed

to find the integrals of polynomials.

## Example 1

As you've seen above,

the coefficients and the plus minus signs

are not affected by the integration.

So focus on the variable parts

and integrate each term.

∫ *x*^{2} *dx* = (1/3)*x*^{3}

∫ *x* *dx* = (1/2)*x*^{2}

∫ 1 *dx* = 1⋅*x*^{1}

And don't forget to add +*C*.

Power rule in integration

So (given) = 2*x*^{3} - *x*^{2} + 5*x* + *C*.

## Example 2

Focus on the variable parts

and integrate each term.

∫ *x*^{4} *dx* = (1/5)*x*^{5}

∫ *x*^{3} *dx* = (1/4)*x*^{4}

∫ *x* *dx* = (1/2)*x*^{2}

∫ 1 *dx* = 1⋅*x*^{1}

Don't forget to add +*C*.

Power rule in integration

So (given) = *x*^{5} - 8*x*^{4} + 7*x*^{2} + 3*x* + *C*.