Problem 25

Question

Find each indefinite integral. $\int(x-1)^{2} d x$

Step-by-Step Solution

Verified

Answer

$\frac{x^3}{3} - x^2 + x + C$

1Step 1: Expand the Expression

Before integrating, expand the expression $(x-1)^2$. This means multiplying $(x-1)$ by itself:\[(x-1)^2 = (x-1)(x-1) = x^2 - 2x + 1\].

2Step 2: Set Up the Integral

Rewrite the integral with the expanded form of the expression from Step 1:\[\int (x^2 - 2x + 1) \, dx\].

3Step 3: Integrate Term by Term

Integrate each term in the expression separately:- The integral of $x^2$ is $\frac{x^3}{3}$.- The integral of $-2x$ is $-x^2$.- The integral of $1$ is $x$.Combine these results:\[\int (x^2 - 2x + 1) \, dx = \frac{x^3}{3} - x^2 + x + C\] where $C$ is the constant of integration.

Key Concepts

Expanding ExpressionsTerm-by-Term IntegrationConstant of Integration

Expanding Expressions

When faced with an integral of a polynomial squared, like $(x-1)^2$, it's crucial to expand the expression before proceeding with integration. Expanding refers to multiplying out the terms in the expression, which simplifies the integration process. In our example, we start with $(x-1)(x-1)$. Breaking this down, we multiply each part term by term:

First multiply: $x \times x = x^2$
Outer multiply: $x \times -1 = -x$
Inner multiply: $-1 \times x = -x$
Last multiply: $-1 \times -1 = 1$

Adding these together, we get $x^2 - 2x + 1$. This expanded form makes the integral straightforward, allowing each term to be addressed separately in the next steps.

Term-by-Term Integration

Once the expression is expanded, the next key step is term-by-term integration. This approach involves integrating each term in the polynomial individually, which simplifies the computational process.Looking at the integral $\int (x^2 - 2x + 1) \, dx$, it's crucial to treat each term separately:

For $x^2$, the integral is $\frac{x^3}{3}$. You increase the power by one and divide by the new power.
The integral of $-2x$ becomes $-x^2$. Here, you follow the same rule by increasing the power and dividing.
Lastly, the integral of a constant like $1$ is $x$.

Combining these results, we can write the integrated form: $\frac{x^3}{3} - x^2 + x$. At this point, don't forget about the constant of integration, denoted by $C$, which captures any constant value lost during differentiation.

Constant of Integration

The final touch to integrating an indefinite integral is ensuring the inclusion of the constant of integration, represented as $C$. This constant is important because when taking the indefinite integral, you are essentially finding a family of functions that could produce the original derivative function.In the context of our example with the integral $\int (x^2 - 2x + 1) \, dx$, we arrive at the resulting function $\frac{x^3}{3} - x^2 + x + C$.Here's why $C$ is essential:

An indefinite integral represents an infinite number of possible antiderivatives.
Each possible solution differs by just a constant.
Including $C$ acknowledges that all those solutions are valid.This detail becomes crucial in calculus, where initial conditions or other information might fix the value of $C$ later on.

Thus, always remember to add $C$ whenever you calculate an indefinite integral to fully represent the solution space.

Problem 25

Problem 26

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Problem 25

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Problem 26

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Problem 26

Find each indefinite integral. $\int\left(x^{-2}-x^{-1}+1-x+x^{2}\right) d x$

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