Donald Johnson

2021-12-30

find the indefinite integral and check the result by differentiation.
$\int \left(x+7\right)dx$

Raymond Foley

Step 1 Solve the indefinite integral
The integral mentioned in the question is,
$\int \left(x+7\right)dx$
Let this indefinite integral be equal to a function y.
Thus,
$y=\int \left(x+7\right)dx$
Now, solve the indefinite integral.
$y=\frac{{x}^{2}}{2}+7x$
Step 2 Check the result by differentiation
The solution of the given indefinite integral is obtained as,
$y=\frac{{x}^{2}}{2}+7x$
Differentiate the given function to see whether it gives back the initial integral.
$\frac{dy}{dx}=\frac{1}{2}\frac{d}{dx}\left({x}^{2}\right)+\frac{d}{dx}\left(7x\right)$
$⇒\frac{dy}{dx}=\frac{1}{2}\left(2x\right)+\left(7\right)$
$⇒\frac{dy}{dx}=x+7$
This is the same as the initial expression. Thus, the solution to the given initial integral is correct.

Edward Patten

It is required to calculate:
$\int \left(x+7\right)dx$
Lets

karton

$\int x+7dx$
Use properties of integrals
$\int xdx+\int 7dx$
Evaluate the integrals
$\frac{{x}^{2}}{2}+7x$
$\frac{{x}^{2}}{2}+7x+C$