dx.jpeg "(12x2-4x+1)dx")
Integral of (12x^2 - 4x + 1) dx
Introduction
In this article, we will discuss the integral of the function (12x^2 - 4x + 1) dx. This is a fundamental concept in calculus and is widely used in various fields such as physics, engineering, and mathematics.
The Integral
The integral of (12x^2 - 4x + 1) dx is given by:
∫(12x^2 - 4x + 1) dx = 4x^3 - 2x^2 + x + C
where C is the constant of integration.
Step-by-Step Solution
To evaluate the integral, we can start by breaking down the function into its individual components:
∫(12x^2 - 4x + 1) dx = ∫12x^2 dx - ∫4x dx + ∫1 dx
Next, we can integrate each component separately:
∫12x^2 dx = 4x^3 + C1 ∫-4x dx = -2x^2 + C2 ∫1 dx = x + C3
Now, we can combine the results:
∫(12x^2 - 4x + 1) dx = 4x^3 - 2x^2 + x + C
where C = C1 + C2 + C3 is the constant of integration.
Conclusion
In this article, we have evaluated the integral of (12x^2 - 4x + 1) dx and obtained the result of 4x^3 - 2x^2 + x + C. This integral is an important concept in calculus and has many applications in various fields.
