Chapter 12, Problem 1RE

To determine

To find: The integral value for ${\displaystyle \int \left(6x+3\right)dx}$.

Answer to Problem 1RE

The integral value for ${\displaystyle \int \left(6x+3\right)dx}$ is $3x^2+3x+C$.

Explanation of Solution

Formula used:

For a constant C,

${\displaystyle \int x^{n}dx}=\frac{x^{n+1}}{n+1}+C,n\ne -1$ (1)

${\displaystyle \int \left[f\left(x\right)\pm g\left(x\right)\right]dx}={\displaystyle \int f\left(x\right)dx}\pm {\displaystyle \int g\left(x\right)dx}$ (2)

Calculation:

Let $f\left(x\right)=6x+3$.

Use the formula (1) and (2) to obtain the integral of $f\left(x\right)$ as follows.

$\begin{array}{c}{\displaystyle \int \left(6x+3\right)dx}={\displaystyle \int \left(6x\right)dx}+{\displaystyle \int \left(3\right)dx}\\ =6{\displaystyle \int xdx}+3{\displaystyle \int dx}\\ =6\left(\frac{x^{1+1}}{1+1}\right)+3x+C\\ =3x^2+3x+C\end{array}$

Thus, the integral value for ${\displaystyle \int \left(6x+3\right)dx}$ is $3x^2+3x+C$.