Indefinite Integration of Polynomials

Indefinite Integration of Polynomials

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

Formula: Constant Multiple Rule in Integration

(integral of k*f(x) dx) = k*(integral of f(x) dx)

The coefficient k is not affected
by the integration.
(Just like the coefficient in differentiation.)

So ∫ kf(x) dx = kf(x) dx.

This formula is needed
to find the integrals of polynomials.

Formula: Sum Rule (and Subtraction Rule) in Integration

(integral of [f(x) +- g(x)] dx) = (integral of f(x) dx) +- (integral of g(x) dx)

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

Find the given indefinite integral. The integral of (6x^2 - 2x + 5) dx

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.

x2 dx = (1/3)x3
x dx = (1/2)x2
∫ 1 dx = 1⋅x1

And don't forget to add +C.

Power rule in integration

So (given) = 2x3 - x2 + 5x + C.

Example 2

Find the given indefinite integral. The integral of (5x^4 - 32x^3 + 14x - 3) dx

Focus on the variable parts
and integrate each term.

x4 dx = (1/5)x5
x3 dx = (1/4)x4
x dx = (1/2)x2
∫ 1 dx = 1⋅x1

Don't forget to add +C.

Power rule in integration

So (given) = x5 - 8x4 + 7x2 + 3x + C.