How to Integrate a Polynomial

Problem

\int \left(6x^2 + 4x - 3\right) dx

Answer

2x^3 + 2x^2 - 3x + C

Integration splits over sums and differences. Integrate one term at a time, then add C to cover every possible constant.

Step-by-step solution

Integrate each term separately.
$\int 6x^2\,dx + \int 4x\,dx - \int 3\,dx$
Why: The integral of a sum is the sum of the integrals.
Apply the power rule to the first term.
$\int 6x^2\,dx = 6 \cdot \frac{x^3}{3} = 2x^3$
Why: Raise the power by one and divide by the new power.
Apply the power rule to the second term and the constant.
$\int 4x\,dx = 2x^2,\quad \int 3\,dx = 3x$
Why: The constant term is 3x, because the derivative of 3x is 3.
Add the constant of integration.
$2x^3 + 2x^2 - 3x + C$
Why: Every antiderivative can differ by a constant.

Forgetting the C. An indefinite integral is a family of functions.
Raising the power but forgetting to divide by the new power.
Integrating the constant -3 as 0. A constant integrates to the constant times x.

Use the same method. Work each problem on paper, then open the answer to check.

$\int \left(3x^2 + 2x\right) dx$

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$x^3 + x^2 + C$
$\int \left(4x^3 - 6x\right) dx$

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$x^4 - 3x^2 + C$
$\int \left(2x + 5\right) dx$

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$x^2 + 5x + C$
$\int \left(9x^2 - 1\right) dx$

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$3x^3 - x + C$
$\int \left(x^3 + x\right) dx$

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$\frac{x^4}{4} + \frac{x^2}{2} + C$