Problems from Direct integrals for the polynomials

Development:

We will apply the following procedure:

• Separate the integral into several integrals (one for every term) and extract the constants out of the integral. $$\int (12x^4+3x^2+5x+3)dx=12\int x^{4}dx+3\int x^{2}dx+5\int xdx+3\int 1dx$$

• Use the formula to obtain the result of the integral of every term and add the results. $$12\int x^{4}dx+3\int x^{2}dx+5\int xdx+3\int 1dx=12\cdot {\displaystyle \frac{x^5}{5}}+3\cdot {\displaystyle \frac{x^3}{3}}+5\cdot {\displaystyle \frac{x^2}{2}}+3x$$

• Add the integration constant to the result. $$\int (12x^4+3x^2+5x+3)dx=12\cdot {\displaystyle \frac{x^5}{5}}+3\cdot {\displaystyle \frac{x^3}{3}}+5\cdot {\displaystyle \frac{x^2}{2}}+3x+C$$

Solution:

${\displaystyle \int (12x^4+3x^2+5x+3)dx=12\cdot {\displaystyle \frac{x^5}{5}}+3\cdot {\displaystyle \frac{x^3}{3}}+5\cdot {\displaystyle \frac{x^2}{2}}+3x+C}$

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