Constant multiples, sums and differences (polynomials)

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

One $C$ at the end is enough.

Examples

1. 01Integrate:$\displaystyle \int (3x^2-6x+1)\,dx$AnswerHide

   Answer$\displaystyle x^3-3x^2+x+C$

   Steps

   1. $\displaystyle \int (3x^2-6x+1)\,dx$$\displaystyle = \frac{3}{3}x^3 - \frac{6}{2}x^2 + 1x + C$
   2. Tidy the coefficients
   3. $\displaystyle \frac{3}{3}x^3 - \frac{6}{2}x^2 + 1x$$\displaystyle = x^3-3x^2+x$
2. 02Integrate$\displaystyle \int x^{2} \, dx$AnswerHide

   Answer$\displaystyle \frac{1}{3}x^{3} + C$
3. 03Integrate:$\displaystyle \int (6x^2+6x)\,dx$AnswerHide

   Answer$\displaystyle 2x^3+3x^2+C$

   Steps

   1. $\displaystyle \int (6x^2+6x)\,dx$$\displaystyle = \frac{6}{3}x^3 + \frac{6}{2}x^2 + C$
   2. Tidy the coefficients
   3. $\displaystyle \frac{6}{3}x^3 + \frac{6}{2}x^2$$\displaystyle = 2x^3+3x^2$
4. 04Integrate:$\displaystyle \int (9x^2-4x+4)\,dx$AnswerHide

   Answer$\displaystyle 3x^3-2x^2+4x+C$

   Steps

   1. $\displaystyle \int (9x^2-4x+4)\,dx$$\displaystyle = \frac{9}{3}x^3 - \frac{4}{2}x^2 + 4x + C$
   2. Tidy the coefficients
   3. $\displaystyle \frac{9}{3}x^3 - \frac{4}{2}x^2 + 4x$$\displaystyle = 3x^3-2x^2+4x$
5. 05Integrate:$\displaystyle \int (6x^2-4x+4)\,dx$AnswerHide

   Answer$\displaystyle 2x^3-2x^2+4x+C$

   Steps

   1. $\displaystyle \int (6x^2-4x+4)\,dx$$\displaystyle = \frac{6}{3}x^3 - \frac{4}{2}x^2 + 4x + C$
   2. Tidy the coefficients
   3. $\displaystyle \frac{6}{3}x^3 - \frac{4}{2}x^2 + 4x$$\displaystyle = 2x^3-2x^2+4x$
6. 06Integrate:$\displaystyle \int (9x^2-2x+5)\,dx$AnswerHide

   Answer$\displaystyle 3x^3-x^2+5x+C$

   Steps

   1. $\displaystyle \int (9x^2-2x+5)\,dx$$\displaystyle = \frac{9}{3}x^3 - \frac{2}{2}x^2 + 5x + C$
   2. Tidy the coefficients
   3. $\displaystyle \frac{9}{3}x^3 - \frac{2}{2}x^2 + 5x$$\displaystyle = 3x^3-x^2+5x$

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