Question

What is $\int{\left( x+7 \right)dx}$?

Answer 1

In this given question, we are about the integration of the function. i.e., $\int{\left( x+7 \right)dx}$. In order to solve the questions, we have to use trigonometric integration formulas to simplify the given integration function and then follow integration formulas or methods to integrate. First, we separate the integral into the sum of two terms, and then we apply the integration formulas according to the given question, and finally we get the required integration.

Complete step by step solution:
The basic formula for integration is taken from the fundamental theorem of calculus
i.e., $\int{f\left( x \right)}dx=F\left( x \right)+C$ where $F'\left( x \right)=f\left( x \right)$
Let us solve the given question,
Given, $\int{\left( x+7 \right)dx}$=?
Let this equation be I which is integral, so we can write this as,
$I=\int{\left( x+7 \right)dx}$……………….. (1)
The fundamental theorem of calculus states that
$\int{f\left( x \right)}dx=F\left( x \right)+C$
Where, $F'\left( x \right)=f\left( x \right)$
Then writing the equation (1) integral into the sum of two terms,
$\Rightarrow I=\int{xdx+}\int{7dx}$…………. (2)
The integration formula for $\int{xdx\text{ is }d\dfrac{{{x}^{2}}}{2}}+C$is obtained by the power rule
and since $\int{cdx}\text{=c}\int{dx}$ then$\int{7dx}\text{ is }7x+C$
And since the integral of a sum is sum of integrals
Then, applying the trigonometric integration properties in equation (2)
$\Rightarrow I=\dfrac{{{x}^{2}}}{2}+7x+C$
$\therefore \int{\left( x+7 \right)dx}=\dfrac{{{x}^{2}}}{2}+7x+C$
Hence, this is the required integration.

Note: When doing indefinite integration, always write +C part after the integration. This +C part indicates the constant part remains after integration and can be understood when you explore it graphically. All the methods for integration should be always remembered so that we can easily choose which method is suitable for solving the particular type of question. We should do all the calculations carefully and explicitly to avoid making errors.