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Chapter 5: Problem 9

\(5-18\) Find the general indefinite integral. $$\int(1-t)\left(2+t^{2}\right) d t$$

Short Answer

Expert verified
The indefinite integral is \(2t + \frac{t^3}{3} - t^2 - \frac{t^4}{4} + C.\)

Step by step solution

01

Expand the Expression

First, expand the expression within the integral \((1-t)(2+t^2) = 1 \times 2 + 1 \times t^2 - t \times 2 - t \times t^2\)which simplifies to \(2 + t^2 - 2t - t^3.\)
02

Rewrite the Integral

Rewrite the integral using the expanded expression from Step 1:\(\int(2 + t^2 - 2t - t^3) \, dt.\)
03

Integrate Term-by-Term

Integrate each term separately:- The integral of 2 with respect to t is \(2t\).- The integral of \(t^2\) is \(\frac{t^3}{3}\).- The integral of \(-2t\) is \(-t^2\).- The integral of \(-t^3\) is \(-\frac{t^4}{4}\).So, the integral becomes:\(2t + \frac{t^3}{3} - t^2 - \frac{t^4}{4} + C,\)where \(C\) is the constant of integration.
04

Write Down the Final Answer

Combine all terms to write the final expression:\(\int(1-t)(2+t^2) \, dt = 2t + \frac{t^3}{3} - t^2 - \frac{t^4}{4} + C.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Techniques
When tackling indefinite integrals, one key strategy is to simplify the expression first by using appropriate integration techniques. In our case, the function \( (1-t)(2+t^2) \) requires expansion before integration.
By expanding, we convert it into a polynomial \( 2 + t^2 - 2t - t^3 \), which is easier to integrate term-by-term. Integration involves finding antiderivatives, which are functions that describe the area under a curve. For polynomial terms:
  • Each term is integrated individually.
  • Use power rule: \( rac{t^{n+1}}{n+1} \) for each \( t^n \) term.
  • Constants like 2 are integrated by simply multiplying them with the variable, \( 2t \).
This straightforward technique helps in managing each segment of the function separately, making complex problems more approachable.
Polynomial Expansion
Polynomial expansion is a fundamental algebraic technique that simplifies expressions for integration. It involves distributing each term in the parenthesis. In the given function \( (1-t)(2+t^2) \), we apply the distributive property: \( (a+b)(c+d) = ac + ad + bc + bd \).
This results in the expanded polynomial \( 2 + t^2 - 2t - t^3 \).Why expand?
  • Easier integration: Expanded forms reveal each term for simple integration.
  • Avoids unnecessary complications during calculus operations.
  • Reduces risk of errors in computation.
It aids in transforming complex expressions into simpler, standard polynomial form, making the application of calculus rules more direct and intuitive.
Constant of Integration
In indefinite integrals, the constant of integration, often represented by \( C \), is crucial yet easy to overlook. This constant emerges because indefinite integrals represent a family of functions, all of which have the same derivative.
Why is it important?
  • Captures all possible vertical shifts of an antiderivative graph.
  • Ensures solutions are comprehensive, not tied to a specific scenario.
  • Crucial for solving differential equations requiring unique solutions.
For example, the integral we solved was expressed as \( 2t + rac{t^3}{3} - t^2 - rac{t^4}{4} + C \), where \( C \) encapsulates all constant values that could be added to the antiderivative without affecting the integrity of the derivative function. It emphasizes the underlying principle that an integral does not have a single solution but a family of solutions.

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