How to evaluate the definite integral of $\int{({{x}^{2}}+1)dx}$ from $[1,2]$ ?

Hint: To solve this question we need to know the concept of definite integral. We are given with the limits of the integral for the given function. For integrating the given function, ${{x}^{2}}+1$ the formula used will be $\dfrac{{{x}^{n+1}}}{n+1}$, only if the function given is ${{x}^{n}}$. So if the function to be integrated is $1$ then power of $x$ will be $0$, resulting in the integration to be $x$.

Complete step-by-step solution:
To solve the integration of the function we should know the formula of the integration. The question is $\int{({{x}^{2}}+1)dx}$ with limit as $[1,2]$, which could be written as
$\int\limits_{1}^{2}{({{x}^{2}}+1)}dx$
The formula used to integrate the formula is ,if the function is ${{x}^{n}}$ then integration of the function becomes $\dfrac{{{x}^{n+1}}}{n+1}$.
$\Rightarrow \int\limits_{1}^{2}{{{x}^{2}}dx+\int\limits_{1}^{2}{1dx}}$
Considering the first integral, on applying the same concept to the function given to us we have $n=2$.
$\Rightarrow \int\limits_{1}^{2}{{{x}^{2}}dx}$
On applying the formula we get
$\Rightarrow \dfrac{{{x}^{2+1}}}{2+1}$
Now, integrating the second function which is $\int\limits_{1}^{2}{1dx}$,
$\int\limits_{1}^{2}{1dx}$
Here $n=0$, on applying the above formula in this function we get:
$\Rightarrow \dfrac{{{x}^{0+1}}}{1}$
So the function after integration is :
$\Rightarrow \left[ \dfrac{{{x}^{2+1}}}{2+1}+~\dfrac{{{x}^{0+1}}}{1} \right]_{1}^{2}$
$\Rightarrow \left[ \dfrac{{{x}^{3}}}{3}+~\dfrac{{{x}^{1}}}{1} \right]_{1}^{2}$
After putting the limits on the integration we the the following expression:
$\Rightarrow \left[ \dfrac{{{2}^{3}}}{3}+\dfrac{2}{1}-\left( \dfrac{1}{3}+\dfrac{1}{1} \right) \right]$
On solving the small bracket first, and then the big bracket we get :
$\Rightarrow \left[ \dfrac{8}{3}-\dfrac{1}{3}+2-1 \right]$
$\Rightarrow \dfrac{7}{3}+1$
$\Rightarrow \dfrac{10}{3}$
$\therefore $ The integration of the function $\int{({{x}^{2}}+1)dx}$ from $[1,2]$ we get $\dfrac{10}{3}$.

Note: To solve this question we need to remember the formulas for integration, so that we do not make mistakes while writing the formulas. If the function that needs to be integrated is a big or complicated one, then we can divide that function in parts and integrate as done in this question. It is done so as to remove or avoid errors. The concept of definite integral is very useful and has a large number of applications. It is used to find the area of the enclosed figure etc.