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Magazine Chapter 7: Problem 32
Evaluate the following integrals. $$\int_{1}^{2} \frac{2}{t^{3}(t+1)} d t$$
Short Answer
Expert verified
Question: Evaluate the integral \(\displaystyle \int_{1}^{2} \frac{2}{t^3(t+1)} d t\)
Answer: \(\displaystyle -\ln{2} + \frac{3}{4}\)
Step by step solution
01
Partial fraction decomposition
First, we perform a partial fraction decomposition of the expression. We have:
$$\frac{2}{t^3(t+1)} = \frac{A}{t} + \frac{B}{t^2} + \frac{C}{t^3} + \frac{D}{t+1}$$
Multiply both sides by \(\displaystyle t^3(t + 1)\) to get rid of the denominator:
$$2 = A t^2 (t + 1) + B t (t + 1) + C (t + 1) + D t^3$$
02
Find coefficients A, B, C, and D
We need to find the appropriate values for A, B, C, and D. We can plug in convenient values for t:
1. Setting t = 0, we get:
$$2 = 0A + 0B + C + 0D \Rightarrow C = 2$$
2. Setting t = -1, we get:
$$2 = 0A + 0B - 2C + D(-1)^3 \Rightarrow D = 2$$
3. Setting t = 1, we get:
$$2 = A(1)(2) + B(1)(2) + 2(1) + 2(1) \Rightarrow 2A + 2B = -2$$
4. Differentiating the equation once with respect to t, we get:
$$0 = 2 A t (t + 1) + A t^2 + 2 B t + B + 3 C t^2 + D (3 t^2 + 1)$$
Setting t = 0,
$$0 = 0A + 0B + 0C + D(0)^3 + D \Rightarrow D = 0$$
Now, since D = 0 and C = 2, we can plug them back into the equation we got from Step 3 and get:
$$-2 = 2A + 2B \Rightarrow A + B = -1$$
We will solve for A and B, determining their specific values.
03
Solve for A and B
We will now solve for specific values of A and B using the following system of linear equations:
$$\begin{cases} A + B = -1 \\ 2A + 2B = -2 \end{cases}$$
We can easily see that A = -1 and B = 0.
04
Write the decomposed expression
Now, we rewrite the integrand with the coefficients we found:
$$\frac{2}{t^3(t+1)} = \frac{-1}{t} + \frac{0}{t^2} + \frac{2}{t^3} + \frac{0}{t+1}$$
Simplify the expression:
$$\frac{2}{t^{3}(t+1)} = -\frac{1}{t} + \frac{2}{t^3}$$
05
Integrate the decomposed expression
Now, we integrate the decomposed expression with respect to t:
$$\int_{1}^{2} \left( -\frac{1}{t} + \frac{2}{t^3} \right) d t = \left[ -\ln{t} -\frac{1}{t^2} \right]_1^2$$
06
Evaluate the integral
Finally, we evaluate the integral using the limits 1 and 2:
$$\left[ -\ln{t} -\frac{1}{t^2} \right]_1^2 = \left( -\ln{2} - \frac{1}{4} \right) - \left( -\ln{1} - 1 \right) = -\ln{2} + \frac{3}{4}$$
The final result is:
$$\int_{1}^{2} \frac{2}{t^3(t+1)} d t = -\ln{2} + \frac{3}{4}$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Partial fraction decomposition
Partial fraction decomposition is a technique used to express a complex rational function as a sum of simpler fractions, making it easier to integrate or differentiate. In essence, we break down a complicated fraction into smaller fractions whose denominators are simpler.
This is particularly useful because integrating simpler fractions is generally more straightforward.In our exercise, we start with the fraction \(\frac{2}{t^3(t+1)}\). The goal is to decompose it into simpler bits like \(\frac{A}{t} + \frac{B}{t^2} + \frac{C}{t^3} + \frac{D}{t+1}\). Each term represents a different part of the original fraction.
To find the coefficients \(A\), \(B\), \(C\), and \(D\), we multiply the given equation by the common denominator to eliminate the fractions. After calculating for specific values of \(t\), we find each coefficient necessary to rewrite the expression. The resulting simplified expression is easier to integrate, transforming our problem from a challenging task into a simpler one.
This is particularly useful because integrating simpler fractions is generally more straightforward.In our exercise, we start with the fraction \(\frac{2}{t^3(t+1)}\). The goal is to decompose it into simpler bits like \(\frac{A}{t} + \frac{B}{t^2} + \frac{C}{t^3} + \frac{D}{t+1}\). Each term represents a different part of the original fraction.
To find the coefficients \(A\), \(B\), \(C\), and \(D\), we multiply the given equation by the common denominator to eliminate the fractions. After calculating for specific values of \(t\), we find each coefficient necessary to rewrite the expression. The resulting simplified expression is easier to integrate, transforming our problem from a challenging task into a simpler one.
Definite integrals
A definite integral is a type of integral that calculates the net area under a curve between fixed limits. Unlike indefinite integrals, which describe a family of functions, definite integrals give a specific numerical value, representing the accumulation of quantities, such as area.When you're evaluating a definite integral, you not only find the antiderivative but also deal with specific upper and lower bounds. These bounds define the interval over which you're considering the area.
In our case, the exercise involves the definite integral \(\int_{1}^{2} \left( -\frac{1}{t} + \frac{2}{t^3} \right) dt\). The numbers 1 and 2 are the limits of integration indicating where you'll evaluate the antiderivative.The solution ultimately computes the difference \([F(2) - F(1)]\), yielding a single number, revealing how two different forces balance over the interval from 1 to 2.
In our case, the exercise involves the definite integral \(\int_{1}^{2} \left( -\frac{1}{t} + \frac{2}{t^3} \right) dt\). The numbers 1 and 2 are the limits of integration indicating where you'll evaluate the antiderivative.The solution ultimately computes the difference \([F(2) - F(1)]\), yielding a single number, revealing how two different forces balance over the interval from 1 to 2.
Improper integrals
Improper integrals are a bit different from regular definite integrals. They can involve functions that have unbounded behavior or infinite intervals. It's like trying to calculate the area when part of your curve stretches infinitely far.
These integrals require additional steps to evaluate, often involving a limit process to handle the infinite aspects.In our specific task provided, the function does not directly involve an improper integral. However, the function \(\frac{2}{t^3(t+1)}\) could potentially lead to improper integrals if the limits of integration were larger, or if it approached an asymptote.
If the behavior at the endpoints was problematic, we would use limits to define the integral properly, ensuring that the infinite behavior is correctly managed. Considering this context helps strengthen understanding for situations where you might face these types of integrals.
These integrals require additional steps to evaluate, often involving a limit process to handle the infinite aspects.In our specific task provided, the function does not directly involve an improper integral. However, the function \(\frac{2}{t^3(t+1)}\) could potentially lead to improper integrals if the limits of integration were larger, or if it approached an asymptote.
If the behavior at the endpoints was problematic, we would use limits to define the integral properly, ensuring that the infinite behavior is correctly managed. Considering this context helps strengthen understanding for situations where you might face these types of integrals.
