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Magazine Chapter 7: Problem 84
Decide whether the statements are true or false. Give an explanation for your answer. The integral \(\int t^{2} e^{3-t} d t\) can be done by parts.
Short Answer
Expert verified
True; it can be solved by using integration by parts.
Step by step solution
01
Identify the Integral Type
The integral given is \( \int t^2 e^{3-t} \, dt \). It consists of a polynomial function \( t^2 \) and an exponential function \( e^{3-t} \). This is a good candidate for integration by parts.
02
Recall Integration by Parts Formula
The integration by parts formula is \( \int u \, dv = uv - \int v \, du \). We need to choose appropriate \( u \) and \( dv \) in our integral to apply this formula.
03
Choose \( u \) and \( dv \)
For our integral \( \int t^2 e^{3-t} \, dt \), we can choose \( u = t^2 \) and \( dv = e^{3-t} \, dt \). This choice is based on the ILATE rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential), where polynomial terms like \( t^2 \) are chosen for \( u \).
04
Differentiate and Integrate
Differentiate \( u \) to get \( du = 2t \, dt \) and integrate \( dv \) to get \( v = -e^{3-t} \) (using substitution for integration). Thus, our \( v \) is the antiderivative of \( dv \).
05
Apply Integration by Parts
Using the integration by parts formula, substitute the values for \( u \), \( dv \), \( du \), and \( v \). The integral becomes \( \int t^2 e^{3-t} \, dt = -t^2 e^{3-t} - \int -2t e^{3-t} \, dt \).
06
Determine if Further Steps are Needed
The resulting integral \( \int 2t e^{3-t} \, dt \) can also be done by parts, repeating similar steps as above, confirming that the original statement is true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral
An integral is a fundamental concept in calculus that represents the accumulation of quantities, like area under a curve. When we talk about definite integrals, we refer to the evaluation over a specified interval, providing a numerical result. Indefinite integrals, like the one in the exercise, represent a family of functions whose derivative gives us the original function. Such integrals oftentimes include a constant of integration, denoted by 'C'.
Understanding when to use different techniques, like integration by parts, is essential. The goal is to simplify or solve the integral by breaking it down into more manageable components.
Understanding when to use different techniques, like integration by parts, is essential. The goal is to simplify or solve the integral by breaking it down into more manageable components.
Polynomial Function
Polynomial functions are expressions involving sums of powers of variables with constant coefficients. They often appear in the form of \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a\) represents coefficients and \(x\) is the variable.
In our given integral \(\int t^2 e^{3-t} \, dt\), \(t^2\) is the polynomial part.
In our given integral \(\int t^2 e^{3-t} \, dt\), \(t^2\) is the polynomial part.
- Polynomials are frequently chosen as the \(u\) term when using integration by parts due to their simplicity in differentiation.
- This choice often results in a simplified form after performing the differentiation part in the integration by parts process.
Exponential Function
Exponential functions involve expressions where the variable is in the exponent, most commonly in the form of \(e^x\), where \(e\) is the mathematical constant approximately equal to 2.718. These functions are crucial in modeling growth processes, such as populations or radioactive decay.
In the integral in question, \(e^{3-t}\) is the exponential component. This term is usually simple to integrate, which makes it an ideal candidate for \(dv\) in integration by parts.
In the integral in question, \(e^{3-t}\) is the exponential component. This term is usually simple to integrate, which makes it an ideal candidate for \(dv\) in integration by parts.
- Exponential functions often don't change drastically when differentiated or integrated, maintaining an exponential form.
- When solving by parts, integrating \(dv\) for an exponential function remains straightforward, simplifying part of the overall integral strategy.
Integration Techniques
Integration techniques are methods used to solve integrals, particularly when they don't fit basic forms. These include fundamental approaches like substitution, integration by parts, and partial fractions.
In this exercise, integration by parts is the technique used. This method is a version of the product rule for differentiation and is outlined by the formula: \[ \int u \, dv = uv - \int v \, du \].
In this exercise, integration by parts is the technique used. This method is a version of the product rule for differentiation and is outlined by the formula: \[ \int u \, dv = uv - \int v \, du \].
- Integration by parts is typically applied to integrals involving products of functions, like a polynomial and an exponential function.
- The choice of \(u\) and \(dv\) is crucial, with mnemonic devices like ILATE (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential) assisting in optimal selection.
- Multiple iterations might be necessary if the resulting integral after the first application of parts still requires further simplification, as seen in the given exercise.
