Integral of 3x^2-√5x+2 | What is integral of ∫(3x^2-√5x+2).dx?

Integral of 3x^2-√5x+2 | What is integral of ∫(3x^2-√5x+2).dx?

Integral of 3x² - √5x + 2: A Step-by-Step Analysis

The process of integration is fundamental in mathematics, particularly in calculus. This article focuses on a specific polynomial function and guides you through the integration process. We will find the integral of the function ( f(x) = 3x^2 - \sqrt{5}x + 2 ).

Understanding the Function

The function we are considering is a polynomial of degree 2. Polynomials are straightforward to integrate, thanks to established rules that govern the process. In this case, we have three terms to integrate:

( 3x^2 )

( -\sqrt{5}x )

( +2 )

Integral Definition

The integral of a function ( f(x) ) can be expressed as:

$\int f(x) , dx$

We seek to determine ( \int (3x^2 - \sqrt{5}x + 2) , dx ).

Step-by-Step Integration

Let’s integrate each term one by one.

Integrating ( 3x^2 ):

The integral of ( x^n ) is given by:

$\int x^n , dx = \frac{x^{n+1}}{n+1} + C

For our term ( 3x^2 ):

$\int 3x^2 , dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3$

Integrating ( -\sqrt{5}x ):

Using the same rule for integration:

$\int -\sqrt{5}x , dx = -\sqrt{5} \cdot \frac{x^{1+1}}{1+1} = -\sqrt{5} \cdot \frac{x^2}{2} = -\frac{\sqrt{5}}{2} x^2$

Integrating ( +2 ):

The integral of a constant ( a ) is:

$\int a , dx = ax + C$

Therefore, for ( +2 ):

$\int 2 , dx = 2x

Combining the Results

Now that we have integrated each term, we can combine the results:

$\int (3x^2 - \sqrt{5}x + 2) , dx = x^3 - \frac{\sqrt{5}}{2} x^2 + 2x + C$

Where ( C ) is the constant of integration, which represents an arbitrary constant that can arise in indefinite integrals.

Final Result

Thus, the integral of the function ( 3x^2 - \sqrt{5}x + 2 ) is:

$\int (3x^2 - \sqrt{5}x + 2) , dx = x^3 - \frac{\sqrt{5}}{2} x^2 + 2x + C$

Conclusion

Integrating polynomial functions like ( 3x^2 - \sqrt{5}x + 2 ) allows us to find areas under curves and solve various mathematical problems. Understanding the rules of integration enables one to tackle more complex functions in calculus. This integral example showcases the foundational principles that are essential for mastering the subject.