Integrals: the basics

Constant multiples, sums and differences (polynomials)

To integrate a polynomial, take constants outside and apply the ∫xn dx\int x^n\,dx formula term by term.

∫kf dx=k∫f dx,∫(f±g) dx=∫f dx±∫g dx\begin{gathered} \int kf\,dx=k\int f\,dx, \\[4pt] \int (f\pm g)\,dx=\int f\,dx\pm\int g\,dx \end{gathered}

As with differentiation, constant multiples come outside and sums and differences are integrated term by term. For a polynomial, apply the ∫xn dx\int x^n\,dx formula to each term.

∫(6x2−4x+1) dx=2x3−2x2+x+C\int (6x^2-4x+1)\,dx=2x^3-2x^2+x+C

One CC at the end is enough.

Examples

  1. 01Integrate:∫(3x2−6x+1) dx\displaystyle \int (3x^2-6x+1)\,dxAnswerHide
    Answerx3−3x2+x+C\displaystyle x^3-3x^2+x+C
  2. 02Integrate:∫(6x2+6x) dx\displaystyle \int (6x^2+6x)\,dxAnswerHide
    Answer2x3+3x2+C\displaystyle 2x^3+3x^2+C
  3. 03Integrate:∫(9x2−4x+4) dx\displaystyle \int (9x^2-4x+4)\,dxAnswerHide
    Answer3x3−2x2+4x+C\displaystyle 3x^3-2x^2+4x+C
  4. 04Integrate∫x2 dx\displaystyle \int x^{2} \, dxAnswerHide
    Answer13x3+C\displaystyle \frac{1}{3}x^{3} + C
  5. 05Integrate∫x3 dx\displaystyle \int x^{3} \, dxAnswerHide
    Answer14x4+C\displaystyle \frac{1}{4}x^{4} + C
  6. 06Integrate∫x5 dx\displaystyle \int x^{5} \, dxAnswerHide
    Answer16x6+C\displaystyle \frac{1}{6}x^{6} + C

FAQ

Q1Do I add C to every term?Show answerHide
A
One C at the end is enough. The constants from each term are combined into a single C.

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The formula for ∫x^n dx
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Published 2026-09-03 · Updated 2026-09-05