Problem 3 - Single Variable Calculus

Question

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(3-2 x+x^{2}\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is: \[ \int(3-2x+x^{2})\, dx = \frac{x^3}{3} - x^2 + 3x + C \]

Step 1: Integrate each term separately

We will integrate each term in the polynomial separately: \(3\), \(-2x\), and \(x^2\).

Step 2: Integrate the constant term \(3\)

The indefinite integral of a constant is simply the constant multiplied by the independent variable: \[ \int 3 \, dx = 3x + C_1 \] where \(C_1\) is the constant of integration for this term.

Step 3: Integrate the linear term \(-2x\)

For this term, we will apply the power rule of integration, which states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C_2\), where \(n \neq -1\). In this case, \(n = 1\): \[ \int (-2x) \, dx = -2\int x^1 \, dx = -2\frac{x^{1+1}}{1+1} + C_2 = -x^2 + C_2 \] where \(C_2\) is the constant of integration for this term.

Step 4: Integrate the quadratic term \(x^2\)

Again, applying the power rule of integration with \(n = 2\): \[ \int x^2 \, dx = \frac{x^{2+1}}{2+1} + C_3 = \frac{x^3}{3} + C_3 \] where \(C_3\) is the constant of integration for this term.

Step 5: Combine the integrals

We will now combine the results from Steps 2, 3, and 4 to find the indefinite integral of the given polynomial function: \[ \int (3 -2x + x^2) \, dx = (3x + C_1) + (-x^2 + C_2) + \left(\frac{x^3}{3} + C_3\right) = \frac{x^3}{3} - x^2 + 3x + C \] where \(C = C_1 + C_2 + C_3\) is the constant of integration for the entire polynomial.

Step 6: Differentiate the indefinite integral

Now we will differentiate the indefinite integral we just found to check our answer: \[ \frac{d}{dx}\left(\frac{x^3}{3} - x^2 + 3x + C\right) = \frac{3x^2}{3} -2x + 3 = x^2 - 2x + 3 \] Our differentiated result is the same as the original function, confirming that our integration is correct. Therefore, the indefinite integral of the given function is: \[ \int(3-2x+x^{2})\, dx = \frac{x^3}{3} - x^2 + 3x + C \]

Key Concepts

Power Rule of Integration

The power rule of integration is one of the most fundamental tools for finding indefinite integrals of polynomial functions. This rule helps us integrate expressions of the form \(x^n\), where \(n\) is a real number. The power rule states:

• \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)

Here, \(C\) represents the constant of integration.
To apply this rule, simply increase the exponent by one, then divide by the newly increased exponent. For example, applying this to \(x^2\), we increase the exponent to 3 and then divide by 3:

• \(\int x^2 \, dx = \frac{x^3}{3} + C\)

This rule is particularly useful in our exercise to integrate the term \(x^2\), leading us to the result \(\frac{x^3}{3}\). By understanding and applying the power rule, we can handle more complex polynomial integrations effectively.

Constant of Integration

In indefinite integrals, the constant of integration \(C\) is a crucial component that arises because there are infinitely many antiderivatives for a given function. When you integrate a function, the result can have a variety of forms shifted vertically by different constants.
Each time we integrate a term separately, we add a constant of integration, like \(C_1, C_2, C_3\), to every distinct term. This reflects the fact that the indefinite integral includes all possible antiderivatives. In practice, we only need one constant after combining all terms:

• When we combine them: \(C = C_1 + C_2 + C_3\)

This ensures the indefinite integral reflects all possible antiderivatives of the expression. Recognizing the role of the constant is vital, as it accounts for the infinite family of curves that can solve the integral.

Differentiation Check

Performing a differentiation check is a smart way to verify that your calculated indefinite integral is correct. You simply differentiate your result and see if you arrive back at the original function you intended to integrate. For instance, we worked through the integral \(\int (3 - 2x + x^2) \, dx\). After calculating:

• \(\frac{x^3}{3} - x^2 + 3x + C\)

We differentiate this expression:

• \(\frac{d}{dx}\left(\frac{x^3}{3} - x^2 + 3x + C\right) = x^2 - 2x + 3\)

This returns to our original polynomial \(3 - 2x + x^2\). This step proves that our integration was performed correctly, serving as a powerful verification method. Conducting a differentiation check should always be considered to confirm your results are precise and accurate.