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Magazine Chapter 11: Problem 49
Find the indefinite integral. $$\int \frac{(\sqrt{x}-1)^{2}}{x^{2}} d x$$
Short Answer
Expert verified
The indefinite integral of the given function is: \[ - \frac{1}{2}x^{-2} + C \]
Step by step solution
01
Identify a suitable substitution
The integral has a composite function - a function within a function: \((\sqrt{x}-1)^{2}\). We can choose to substitute \( u = \sqrt{x} - 1\), which simplifies the expression.
02
Calculate the differential
To make the substitution, we need to find the differential \(du\). To do this, we first find the derivative of \(u\) with respect to \(x\), then multiply by \(dx: \qquad d u = \frac{d u}{d x} d x \).
So, \(u = \sqrt{x} - 1 \) or \( x = (u + 1)^{2}\). Now, we calculate the derivative \(\frac{d u}{d x} \), which is \(\frac{d u}{d x} = \frac{1}{2\sqrt{x}} \)
Now, multiply by \( dx \) to get \(d u = \frac{1}{2\sqrt{x}} d x\)
03
Substitute
Now, let's substitute into the integrand:
$$\int \frac{(\sqrt{x}-1)^{2}}{x^{2}} d x =\int \frac{u^{2}}{((u+1)^{2})^{2}} \cdot \frac{1}{2\sqrt{x}} d x $$
Since \( x = (u + 1)^{2}\), we have \(\sqrt{x} = u+1\). Thus, the differential of \(x\) is:
$$du = \frac{1}{2(u+1)} dx$$
Multiplying both sides by \(2(u+1)\):
$$2(u+1) du=dx$$
Now substituting on the integral:
$$\int \frac{u^{2}}{((u+1)^{2})^{2}} \cdot \frac{1}{2\sqrt{x}} d x =\int \frac{u^{2}}{(u+1)^{4}} \cdot 2(u+1) d u $$
This simplifies to:
$$\int\frac{2u^3}{(u+1)^3} du$$
04
Integrate
Now we can find the integral with respect to u:
$$\int\frac{2u^3}{(u+1)^3} du$$
Now, notice that:
\(\frac{d}{du}((u+1)^{-2})=(-2)(u+1)^{-3}\)
This suggests to make another substitution to integrate:
Let \(v = (u + 1)^{-2}\Rightarrow dv = -2(u + 1)^{-3}du\)
So, we have:
$$=\int-1 \cdot v d v$$
Integrating this expression, we find:
$$=- \frac{1}{2}v^{2} + C$$.
05
Reverse substitutions
Now we reverse our substitutions, first replacing v then x:
\begin{align*}
- \frac{1}{2}v^{2} +C &= -\frac{1}{2}(u+1)^{-4} + C \\
&= -\frac{1}{2}(\sqrt{x} - 1 + 1)^{-4} + C \\
&= -\frac{1}{2}(\sqrt{x})^{-4} + C.
\end{align*}
So, the indefinite integral is:
$$- \frac{1}{2}x^{-2} + C$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Composite Function
A composite function is essentially a combination of two or more functions where the output of one function becomes the input of another. In this exercise, the composite function is \((\sqrt{x}-1)^2\), where the inner function \(\sqrt{x}-1\) is nested within the squaring function. This creates layers of complexity when solving integrals, hence understanding composite functions is crucial. For example, if \(f(x) = \sqrt{x}\) and \(g(x) = x - 1\), then \(f(g(x)) = \sqrt{x} - 1\). This mixture can make direct integration tricky without using a method like substitution.
Substitution Method
The substitution method is a powerful tool used to simplify integrals involving composite functions. It involves replacing a part of the integral with a new variable to make the expression easier to integrate.
In our exercise, we use \(u = \sqrt{x} - 1\), turning a complex expression into something more manageable.
Here鈥檚 how it works:
In our exercise, we use \(u = \sqrt{x} - 1\), turning a complex expression into something more manageable.
Here鈥檚 how it works:
- Identify a part of the integrand that complicates direct integration.
- Replace it with a new variable, like \(u\).
- Find the differential \(du\) by differentiating the substitution expression.
- Substitute back into the integral, simplifying it.
Differential Calculus
Differential calculus is all about understanding how functions change, using derivatives. In the substitution method, we need derivatives to find differentials, \(du\).
For instance, when we set \(u = \sqrt{x} - 1\):
For instance, when we set \(u = \sqrt{x} - 1\):
- We differentiate to find \(\frac{d u}{d x} = \frac{1}{2\sqrt{x}}\).
- Multiplying by \(dx\) gives us \(du = \frac{1}{2\sqrt{x}} dx\).
Integral Calculus
Integral calculus focuses on finding the integral or antiderivative of functions, which is essentially the opposite of differentiation. An indefinite integral, like the one in our exercise, represents a family of functions and includes a constant of integration, \(C\).
In our example, after simplifying the integral using substitution, we found: \[ \int \frac{2u^3}{(u+1)^3} \ du \] The integration process is about finding a function whose derivative would be the integrand.
After integrating, we must carefully backtrack through our substitutions to express the result in terms of the original variable, \(x\). This gives us the final indefinite integral, complete with the integration constant. Understanding how to reverse substitutions and adjust for different constants is vital in solving these types of problems.
In our example, after simplifying the integral using substitution, we found: \[ \int \frac{2u^3}{(u+1)^3} \ du \] The integration process is about finding a function whose derivative would be the integrand.
After integrating, we must carefully backtrack through our substitutions to express the result in terms of the original variable, \(x\). This gives us the final indefinite integral, complete with the integration constant. Understanding how to reverse substitutions and adjust for different constants is vital in solving these types of problems.
