Integrating Complex Expressions: A Step-by-Step Guide to Solving ( int (x-1)^2 , dx )

Understanding Integration and Antiderivatives

Integration is a fundamental concept in calculus, and it is often used to find the area under a curve, among other applications. An antiderivative, or indefinite integral, reverses the process of differentiation, allowing us to find the original function from its derivative. This article will guide you through the process of integrating a complex expression, specifically ( (x-1)^2 ), and demonstrate how to find its antiderivative.

Step 1: Simplifying the Expression

Before we proceed to integration, it is often helpful to simplify the expression. For ( (x-1)^2 ), we start by expanding it:

Multiplication and Expansion

The expression ( (x-1)^2 ) can be expanded using the binomial theorem or simply by multiplying it out:

(x-1)(x-1) x^2 - 2x 1

This simplifies the original expression to a polynomial form, making it easier to integrate.

Step 2: Setting Up the Integral

Now that we have simplified our expression, we can set up the integral:

[ int (x^2 - 2x 1) , dx ]

This integral can be broken down into separate integrals for each term:

[ int x^2 , dx - 2 int x , dx int 1 , dx ]

Step 3: Integrating Each Term

We will integrate each term separately:

Integrating ( x^2 )

[ int x^2 , dx frac{x^3}{3} C_1 ]

Integrating ( -2x )

[ -2 int x , dx -2 left( frac{x^2}{2} right) C_2 -x^2 C_2 ]

Integrating ( 1 )

[ int 1 , dx x C_3 ]

Combining these results, we get:

[ frac{x^3}{3} - x^2 x C ]

where ( C ) is the constant of integration, which represents the sum of ( C_1 C_2 C_3 ).

Step 4: Verifying the Antiderivative

It is always a good practice to check the result by differentiating it. If we differentiate ( frac{x^3}{3} - x^2 x C ), we should get back to the original function:

Verification Process

[ frac{d}{dx} left( frac{x^3}{3} - x^2 x C right) x^2 - 2x 1 ]

This confirms that our antiderivative is correct.

Conclusion

Understanding how to integrate expressions like ( (x-1)^2 ) involves several steps, including simplifying the expression, setting up the integral, integrating each term, and verifying the result. This process not only helps in solving specific problems but also builds a strong foundation in calculus. Whether you are a student or a professional, mastering these techniques is essential for advancing in the field of mathematics and its applications.

Additional Resources

For further practice and in-depth understanding, you might want to explore more resources such as online calculators, textbooks, and video tutorials on integration and antiderivatives.