Compute the indefinite integral (or antiderivative function) of the function $$12x^4+3x^2+5x+3$$, that is, compute $$$ \int (12x^4+3x^2+5x+3) \ dx$$$
See development and solution
Development:
We will apply the following procedure:
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Separate the integral into several integrals (one for every term) and extract the constants out of the integral. $$$ \int (12x^4+3x^2+5x+3) \ dx = 12 \int x^4 \ dx+3 \int x^2 \ dx+5 \int x \ dx+3\int 1 \ dx$$$
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Use the formula to obtain the result of the integral of every term and add the results. $$$12 \int x^4 \ dx+3 \int x^2 \ dx+5 \int x \ dx+3\int 1 \ dx =12\cdot\dfrac{x^5}{5}+3\cdot\dfrac{x^3}{3}+5\cdot\dfrac{x^2}{2}+3x$$$
- Add the integration constant to the result. $$$ \int (12x^4+3x^2+5x+3) \ dx= 12\cdot\dfrac{x^5}{5}+3\cdot\dfrac{x^3}{3}+5\cdot\dfrac{x^2}{2}+3x + C$$$
Solution:
$$\displaystyle\int (12x^4+3x^2+5x+3) \ dx= 12\cdot\dfrac{x^5}{5}+3\cdot\dfrac{x^3}{3}+5\cdot\dfrac{x^2}{2}+3x + C$$