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Magazine Chapter 5: Problem 11
Calculate. $$\int \frac{s}{\left(1+s^{2}\right)^{3}} d s$$
Short Answer
Expert verified
\(-\frac{1}{4(1+s^2)^2} + C\)
Step by step solution
01
Substitute Inner Function
Set a variable \(u = 1 + s^{2}\). Then differentiate \(u\) with respect to \(s\), we get \(du/ds = 2s\), or in differential form, \(du = 2s ds\).
02
Change of Variables in Integral
Substitute \(u\) and \(du\) in the original integral, we get \(\frac{1}{2} \int \frac{1}{u^3} du\). The fraction \(\frac{1}{2}\) is because \(s ds = \frac{du}{2}\) from the \(du\) equation.
03
Integrate with Respect to the New Variable
Now evaluate the new integral. The integral of \(\frac{1}{u^3}\) with respect to \(u\) is \(-\frac{1}{2u^2}\). Then add the constant of integration \(C\). Therefore, the result is \(\frac{1}{2}(-\frac{1}{2u^2}) + C = -\frac{1}{4u^2} + C\).
04
Substitute Back the Original Variable
Substitute \(u = 1 + s^2\) back into the result. The final result is \(-\frac{1}{4(1+s^2)^2} + C\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful technique used in integral calculus to simplify complex integrals by transforming them into an easier form. It involves substituting a part of the integral with a new variable. This method not only simplifies the integration process but also aids in tackling complex functions that are hard to integrate directly.
Here's how you can understand this method:
Here's how you can understand this method:
- Identify the Inner Function: Look for a function within the integral that can be substituted to make integration easier. In this exercise, we identified the function as \(1 + s^2\), and we called it \(u\).
- Derive the New Variables: Once you have \(u\), differentiate it with respect to the original variable \(s\). This gives you \(du = 2s \, ds\), which helps in replacing \(ds\) in the integral.
- Substitution in Integral: Replace all instances of the original variable with the new variable. In our exercise, this led to transforming the integral into \(\frac{1}{2}\int \frac{1}{u^3} \, du\).
Definite Integral
Definite integral is a fundamental concept in calculus used to find the total accumulation of a quantity, which is often represented as the area under a curve. Unlike indefinite integrals, which result in a general form plus a constant, definite integrals result in an exact value.
To compute definite integrals, follow these steps:
To compute definite integrals, follow these steps:
- Change of Limits: After substituting, adjust the limits of integration to match the new variable. In our case, although not explicitly needed, it's a key step in definite integrals.
- Evaluate the Integral: Calculate the antiderivative using the new variable and then apply the limits of integration by finding the difference between the upper and lower bounds.
- Substitute Back: Once you have the result, substitute back to the original variable to get a clear understanding of the context.
Antiderivative
An antiderivative is the opposite of a derivative. It represents a function whose derivative is the given function. Finding the antiderivative is the essence of solving integration problems, as it allows the determination of a general form of the original function.
Focusing on this exercise, we can break down the process:
Focusing on this exercise, we can break down the process:
- Find the Antiderivative: In the context given, we needed to evaluate \(\int \frac{1}{u^3} \, du\). Here, the antiderivative was found to be \(-\frac{1}{2u^2}\).
- Add Constant of Integration: After obtaining the antiderivative, we include a constant \(C\) to indicate the family of all possible solutions since integration can introduce arbitrary constants.
- Revert Substitution: Finally, the antiderivative must be expressed in terms of the original variable \(s\). Thus, replacing \(u\) with \(1 + s^2\), the antiderivative becomes \(-\frac{1}{4(1+s^2)^2} + C\).
