** Integral of 2x - 1/2x + 3 with respect to x **
In this article, we will evaluate the indefinite integral of the function 2x - 1/2x + 3 with respect to x.
The Given Function
The function is:
$2x - \frac{1}{2}x + 3$
Splitting the Function
To evaluate the integral, we can split the function into three separate terms:
$2x - \frac{1}{2}x + 3 = 2x - \frac{1}{2}x + 3$
Evaluating the Integral
Now, we can evaluate the integral of each term separately:
Integral of 2x
The integral of 2x is:
$\int 2x dx = x^2 + C$
where C is the constant of integration.
Integral of -1/2x
The integral of -1/2x is:
$\int -\frac{1}{2}x dx = -\frac{1}{4}x^2 + C$
Integral of 3
The integral of 3 is:
$\int 3 dx = 3x + C$
Combining the Results
Now, we can combine the results of each integral to get the final answer:
$\int (2x - \frac{1}{2}x + 3) dx = x^2 - \frac{1}{4}x^2 + 3x + C$
Simplifying the expression, we get:
$\int (2x - \frac{1}{2}x + 3) dx = \frac{7}{4}x^2 + 3x + C$
And that's the final answer!
