Question

How‌ ‌do‌ ‌you‌ ‌evaluate‌ ‌the‌ ‌indefinite‌ ‌integral‌ ‌$\int{\left(‌ ‌{{x}^{2}}-x+5‌ ‌\right)dx}$?‌

Answer 1

To solve the given integration problem, we should know some of the properties of the integration and integration of some of the functions. The property of integration we should know is that the integration can be separated over the addition of functions, that is if the given expression has more than one function in it, we can separate the integral for each function.
The functions whose integration, we should know are ${{x}^{n}}$, and constant function (k), their integrations are $\dfrac{{{x}^{n+1}}}{n+1}\And kx$respectively.

Complete step-by-step solution:
We are asked to evaluate the indefinite integral $\int{\left( {{x}^{2}}-x+5 \right)dx}$. The given expression to integrate has the addition of three separate functions. We know that the integration can be split on the addition, thus we can also express the given integral as
$\int{\left( {{x}^{2}}-x+5 \right)dx}=\int{{{x}^{2}}dx}-\int{xdx}+\int{5dx}$
To find the integration of the given expression, we need to evaluate these three integrals separately and add them.
We know that the integration of functions of the form ${{x}^{n}}$ is $\dfrac{{{x}^{n+1}}}{n+1}$ respectively. Thus, for ${{x}^{2}}$, we have $n=2$. The integration of ${{x}^{2}}$ is $\dfrac{{{x}^{2+1}}}{2+1}=\dfrac{{{x}^{3}}}{3}$. For $x$, we have $n=1$ thus the integration of $x$ is $\dfrac{{{x}^{1+1}}}{1+1}=\dfrac{{{x}^{2}}}{2}$. As 5 is a constant, its integration would be $5x$.
$\int{\left( {{x}^{2}}-x+5 \right)dx}=\int{{{x}^{2}}dx}-\int{xdx}+\int{5dx}$
Substituting the integration of the above functions, we get
$\int{\left( {{x}^{2}}-x+5 \right)dx}=\dfrac{{{x}^{3}}}{3}-\dfrac{{{x}^{2}}}{2}+5x+C$
Here C is the constant of integration.

Note: To solve problems based on integration, we should know the integration of different functions. Also, it should be remembered that it is very important to write the constant of integration with the final answer while solving the indefinite integration problems, otherwise the answer can become incorrect.