25bb9963-221b-448f-a069-1c730936abce37939ca5-4b68-453d-b4e1-5585bdd24634bf9c8cfb-3123-438a-a941-8945554badac4.4.2.1<p>Compute the indefinite integral (or antiderivative function) of the function $$12x^4+3x^2+5x+3$$, that is, compute $$$ \int (12x^4+3x^2+5x+3) \ dx$$$</p><p>We will apply the following procedure:</p> <ul> <li> <p>Separate the integral into several integrals (one for every term) and extract the constants out of the integral. $$$ \int (12x^4+3x^2+5x+3) \ dx = 12 \int x^4 \ dx+3 \int x^2 \ dx+5 \int x \ dx+3\int 1 \ dx$$$</p> </li> <li> <p>Use the formula to obtain the result of the integral of every term and add the results. $$$12 \int x^4 \ dx+3 \int x^2 \ dx+5 \int x \ dx+3\int 1 \ dx =12\cdot\dfrac{x^5}{5}+3\cdot\dfrac{x^3}{3}+5\cdot\dfrac{x^2}{2}+3x$$$</p> </li> <li>Add the integration constant to the result. $$$ \int (12x^4+3x^2+5x+3) \ dx= 12\cdot\dfrac{x^5}{5}+3\cdot\dfrac{x^3}{3}+5\cdot\dfrac{x^2}{2}+3x + C$$$</li> </ul><p>$$\displaystyle\int (12x^4+3x^2+5x+3) \ dx= 12\cdot\dfrac{x^5}{5}+3\cdot\dfrac{x^3}{3}+5\cdot\dfrac{x^2}{2}+3x + C$$</p>7/18/16, 11:44 AM7/18/16, 11:44 AM