Problem 32
Question
Use a symbolic integration utility to find the indefinite integral. $$ \int\left(1+\frac{1}{t}\right)^{3}\left(\frac{1}{t^{2}}\right) d t $$
Step-by-Step Solution
Verified Answer
The indefinite integral of the function \(\int \frac{e^{1 / \sqrt{x}}}{x^{3 / 2}} d x\) is equal to \(-4/5 \cdot e^{1 / \sqrt{x}} \cdot x^{1 / 2} - 2 / 5 \cdot e^{1 / \sqrt{x}} \cdot x^{3 / 2}\).
1Step 1: Substitution
Begin by substituting a new variable for the exponent in the numerator. Let \(u = 1 / \sqrt{x}\) or \(u^2 = 1 / x\). So, \(2u du = -1 / x^2\, dx\). Then \(dx = -2u x^2\, du = -2u / u^4\, du = -2u^3\, du\). Substitute these back into the integral, the integral now becomes \(-2 \int e^{u} u^3 du\).
2Step 2: Application of Integration by Parts
Apply the integration by parts formula, \(\int u v dx = u \int v dx - \int(u' (\int v dx)) dx\). Let \(u = u^3\) and \(dv = e^u du\). Hence, \(du = 3u^2 du\) and \(v = e^u\). Now, apply the integration by parts formula.
3Step 3: Simplification
Simplify the result from step 2, and express the integral in terms of \(x\) by substituting \(u\) with \(1 / \sqrt{x}\).
4Step 4: Final Solution
Observe that one of the integral parts of the result still remains the same as the initial integral form. Hence, complete the calculation by moving the integral to the left side and factor out the common integral on both sides.
Key Concepts
Indefinite IntegralSubstitution MethodIntegration by PartsExponential Functions
Indefinite Integral
An indefinite integral is like finding the opposite of a derivative. It gives us a family of functions whose derivatives equal the integrand. In other words, when we integrate, we're searching for a function that, when differentiated, returns the original expression under the integral.
When solving an indefinite integral like \[\int \frac{e^{1 / \sqrt{x}}}{x^{3 / 2}} dx\]we find an antiderivative rather than a specific numeric answer. Don’t forget, the indefinite integral includes a constant of integration, often represented by "C." This constant represents any constant value that does not change the derivative.
The real art in solving indefinite integrals is using the right techniques to simplify and evaluate them. Techniques like substitution or integration by parts are common tools to handle these tasks.
When solving an indefinite integral like \[\int \frac{e^{1 / \sqrt{x}}}{x^{3 / 2}} dx\]we find an antiderivative rather than a specific numeric answer. Don’t forget, the indefinite integral includes a constant of integration, often represented by "C." This constant represents any constant value that does not change the derivative.
The real art in solving indefinite integrals is using the right techniques to simplify and evaluate them. Techniques like substitution or integration by parts are common tools to handle these tasks.
Substitution Method
The substitution method is a clever technique for simplifying integrals. It involves changing variables to make an integral easier to evaluate. In our problem, this method transforms a complex integral into a simpler one.
When dealing with:\[\int \frac{e^{1 / \sqrt{x}}}{x^{3 / 2}} dx\]we can let \( u = \frac{1}{\sqrt{x}} \) or equivalently \( u^2 = \frac{1}{x} \). Through this substitution, the integral is rewritten in terms of a new variable \( u \), making it easier to handle.
Key steps include:
When dealing with:\[\int \frac{e^{1 / \sqrt{x}}}{x^{3 / 2}} dx\]we can let \( u = \frac{1}{\sqrt{x}} \) or equivalently \( u^2 = \frac{1}{x} \). Through this substitution, the integral is rewritten in terms of a new variable \( u \), making it easier to handle.
Key steps include:
- Identify a substitution that simplifies the integrand.
- Find the differential \( dx \) in terms of \( du \).
- Rewrite and solve the integral in terms of the new variable.
Integration by Parts
Integration by parts is a technique derived from the product rule for differentiation. It's used when integrating products of functions, essentially swapping complexity within an integral to make it solvable.
The formula is:\[\int u \, dv = u \int v \, dx - \int (u' \int v \, dx) \, dx\]In the problem, after substitution, we encounter:\[-2 \int e^{u} u^3 du\]where we let \( u = u^3 \) and \( dv = e^u \, du \). This means \( du = 3u^2 \, du \) and \( v = e^u \).
Steps to apply it effectively:
The formula is:\[\int u \, dv = u \int v \, dx - \int (u' \int v \, dx) \, dx\]In the problem, after substitution, we encounter:\[-2 \int e^{u} u^3 du\]where we let \( u = u^3 \) and \( dv = e^u \, du \). This means \( du = 3u^2 \, du \) and \( v = e^u \).
Steps to apply it effectively:
- Choose \( u \) and \( dv \) correctly to simplify the integral.
- Compute \( du \) and \( v \).
- Substitute into the formula and solve.
Exponential Functions
Exponential functions are characterized by their base, often the mathematical constant \( e \), raised to a variable exponent. They appear frequently in calculus due to their unique properties.
The function \( e^x \) is special because its derivative and integral are the same: - Derivative: \( \frac{d}{dx} e^x = e^x \)- Integral: \( \int e^x \, dx = e^x + C \).
This self-similarity makes exponential functions rewarding yet challenging when combined with other functions in integrals, like in our problem. Here, \( e^{1/\sqrt{x}} \) involved using substitution to reframe the function in a more palatable form.
Key points to remember:
The function \( e^x \) is special because its derivative and integral are the same: - Derivative: \( \frac{d}{dx} e^x = e^x \)- Integral: \( \int e^x \, dx = e^x + C \).
This self-similarity makes exponential functions rewarding yet challenging when combined with other functions in integrals, like in our problem. Here, \( e^{1/\sqrt{x}} \) involved using substitution to reframe the function in a more palatable form.
Key points to remember:
- Identify the base and exponent - they determine the function's behavior.
- Use substitution to simplify integrals involving exponentials.
- Leverage the similarity of exponential derivatives and integrals for easy solutions.
Other exercises in this chapter
Problem 32
Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Mu
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Find the indefinite integral and check your result by differentiation. $$ \int\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x $$
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Use a computer or programmable calculator to approximate the definite integral using the Midpoint Rule and the Trapezoidal Rule for \(n=4\), \(8,12,16\), and 20
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