Problem 10
Question
Calculate the indefinite integral. $$ \int x(x+1)(x+2) d x $$
Step-by-Step Solution
Verified Answer
The indefinite integral is \( \frac{x^4}{4} + x^3 + x^2 + C \).
1Step 1: Expand the Expression
First, we need to expand the polynomial inside the integral. Start by expanding the two binomials \((x+1)(x+2)\): \[(x+1)(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2\] Now multiply by \(x\):\[x(x^2 + 3x + 2) = x^3 + 3x^2 + 2x\] The integral becomes: \[\int (x^3 + 3x^2 + 2x) \, dx\]
2Step 2: Integrate Term by Term
To find the integral, integrate each term of the polynomial separately.1. Integrate \(x^3\): \[\int x^3 \, dx = \frac{x^4}{4} + C_1\] 2. Integrate \(3x^2\): \[\int 3x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3 + C_2\] 3. Integrate \(2x\): \[\int 2x \, dx = 2 \cdot \frac{x^2}{2} = x^2 + C_3\]
3Step 3: Combine the Results
Combine all the integrals together. Remember the constant of integration \(C\) is arbitrary, so we only need one \(C\): \[\int (x^3 + 3x^2 + 2x) \, dx = \frac{x^4}{4} + x^3 + x^2 + C\]
4Step 4: Finalize the Indefinite Integral
The indefinite integral of the expression is given by: \[\int x(x+1)(x+2) \, dx = \frac{x^4}{4} + x^3 + x^2 + C\] This is the final result of the integration process.
Key Concepts
Polynomial ExpansionIntegrationCalculus StepsConstant of Integration
Polynomial Expansion
Polynomial expansion is a key step in many integration problems. It involves transforming a polynomial expression involving products of binomials into a simpler, expanded form. This is crucial for integration, as it breaks down complex expressions into individual terms which can be more easily managed.
In the given exercise, we have the expression \(x(x+1)(x+2)\). The first step in solving it involves expanding the product \((x+1)(x+2)\). To do this, apply the distributive property: each term in the first binomial is multiplied by each term in the second.
In the given exercise, we have the expression \(x(x+1)(x+2)\). The first step in solving it involves expanding the product \((x+1)(x+2)\). To do this, apply the distributive property: each term in the first binomial is multiplied by each term in the second.
- Expand \((x+1)(x+2)\):
- \(x \times x = x^2\)
- \(x \times 2 = 2x\)
- \(1 \times x = x\)
- \(1 \times 2 = 2\)
- Adding these gives \(x^2 + 3x + 2\).
- Then multiply the entire result by \(x\) to expand completely: \(x(x^2 + 3x + 2) = x^3 + 3x^2 + 2x\).
Integration
Integration is the mathematical process of finding the integral of a function, which can be thought of as the opposite operation to differentiation. In other words, while differentiation deals with the rate of change, integration sums up quantities over an interval to find the accumulated total.
Once we’ve expanded \(x(x+1)(x+2)\) to \(x^3 + 3x^2 + 2x\), the integration becomes more straightforward since each component can be integrated separately. Here's the process of integrating each term:
Once we’ve expanded \(x(x+1)(x+2)\) to \(x^3 + 3x^2 + 2x\), the integration becomes more straightforward since each component can be integrated separately. Here's the process of integrating each term:
- For \(x^3\), the integral is: \[ \int x^3 \, dx = \frac{x^4}{4} + C_1 \]
- For \(3x^2\), the integral is: \[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3 + C_2 \]
- For \(2x\), the integral is: \[ \int 2x \, dx = 2 \cdot \int x \, dx = 2 \cdot \frac{x^2}{2} = x^2 + C_3 \]
Calculus Steps
Calculus steps are structured procedures used to solve problems in calculus. For solving an indefinite integral, follow these fundamental steps:
Step 1: Simplify the Expression
Before integration, simplify or expand the expression. Here, expand the polynomial to \(x^3 + 3x^2 + 2x\).
Step 2: Separate and Integrate Each Term
This means applying the power rule of integration to each term: \(x^n\) becomes \(\frac{x^{n+1}}{n+1}\). Perform this process separately for each term.
Step 3: Combine the Results
Add together all the integrated terms and include the constant of integration, \(C\). This constant arises because indefinite integrals represent a family of functions differing by a constant.
Thus, the complete solution to the problem, after performing these steps, is derived by returning the sum of all the integrated terms along with the constant.
Step 1: Simplify the Expression
Before integration, simplify or expand the expression. Here, expand the polynomial to \(x^3 + 3x^2 + 2x\).
Step 2: Separate and Integrate Each Term
This means applying the power rule of integration to each term: \(x^n\) becomes \(\frac{x^{n+1}}{n+1}\). Perform this process separately for each term.
Step 3: Combine the Results
Add together all the integrated terms and include the constant of integration, \(C\). This constant arises because indefinite integrals represent a family of functions differing by a constant.
Thus, the complete solution to the problem, after performing these steps, is derived by returning the sum of all the integrated terms along with the constant.
Constant of Integration
The constant of integration \(C\) is an essential aspect of indefinite integration. Whenever you integrate a function, you're looking for a whole family of solutions that are only different by a constant, rather than a single function.
Since integration is the reverse of differentiation, and differentiating a constant gives zero, we add \(C\) to express all possible constant vertical shifts of the integral. In this case:
The final integral was:\[\int (x^3 + 3x^2 + 2x) \, dx = \frac{x^4}{4} + x^3 + x^2 + C\]
Since integration is the reverse of differentiation, and differentiating a constant gives zero, we add \(C\) to express all possible constant vertical shifts of the integral. In this case:
The final integral was:\[\int (x^3 + 3x^2 + 2x) \, dx = \frac{x^4}{4} + x^3 + x^2 + C\]
- Each term was calculated individually, but they share a single \(C\) in the final output.
- This \(C\) captures all vertical shifts of the integral, reflecting all potential original functions from which the derivative could have come.
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