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Chapter 7: Problem 47

(a).Make an appropriate \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution), and then confirm that the result is equivalent to that in part (a).$ $$\int e^{-\sqrt{x}} d x$$

Short Answer

Expert verified
The integral \( \int e^{-\sqrt{x}} \, dx \) evaluates to \( -e^{-\sqrt{x}}(\sqrt{x} + 1) + C \).

Step by step solution

01

- Choose a substitution

To solve the integral \( \int e^{-\sqrt{x}} \, dx \), we will use \( u \)-substitution. Let's choose \( u = \sqrt{x} \). This implies that \( x = u^2 \) and \( dx = 2u \, du \).
02

- Rewrite the integral in terms of u

Substitute \( u = \sqrt{x} \) and \( dx = 2u \, du \) into the integral. This gives us \( \int e^{-u} \, 2u \, du \) which simplifies to \( 2 \int u e^{-u} \, du \).
03

- Apply integration by parts

To solve \( \int u e^{-u} \, du \), use integration by parts where \( v = e^{-u} \) and \( dv = -e^{-u} \, du \), and choose \( w = u \) and \( dw = du \). Then the integral becomes \( -u e^{-u} + \int e^{-u} \, du \).
04

- Solve the remaining integral

The remaining integral \( \int e^{-u} \, du \) is standard and equal to \( -e^{-u} \). Thus, the integral in terms of \( u \) is \( -u e^{-u} - e^{-u} = -e^{-u}(u + 1) \).
05

- Substitute back the original variables

Replace \( u \) with \( \sqrt{x} \) to revert to the original variable: \( -e^{-\sqrt{x}}(\sqrt{x} + 1) = -e^{-\sqrt{x}} \sqrt{x} - e^{-\sqrt{x}} + C \), where \( C \) is the constant of integration.
06

- Use CAS to verify the result

Utilize a Computer Algebra System (CAS) to evaluate the integral \( \int e^{-\sqrt{x}} \, dx \) directly. The CAS should output \( -e^{-\sqrt{x}}(\sqrt{x} + 1) + C \), matching the result obtained through substitution and confirming the solutions are equivalent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration by Parts
Integration by Parts is a powerful technique for evaluating integrals, especially when the integrand is a product of functions. It's based on the product rule for differentiation, and the formula is: \[\int u \, dv = uv - \int v \, du \]Here, you select parts of your integrand as \( u \) and \( dv \). You then differentiate \( u \) to find \( du \), and integrate \( dv \) to obtain \( v \).
This process helps simplify an otherwise complex integral. In practice, choose \( u \) as the function whose derivative becomes simpler, and \( dv \) as the remaining part.
  • Step 1: Choose \( u \) and \( dv \).
  • Step 2: Find \( du \) and \( v \).
  • Step 3: Apply the formula.
  • Step 4: Solve the resulting integral.
In our example, we applied integration by parts on the integral \( \int u e^{-u} \, du \), resulting in simpler integrals that are easily solvable. This showcases the strategy's efficiency in breaking down challenging problems.
Integral Evaluation
Integral Evaluation refers to the complete process of computing the value of integrals. This often involves techniques like substitution and integration by parts. In the exercise, u-substitution serves as the initial tool to transform the integral into a more manageable format.
By setting \( u = \sqrt{x} \), we translated our original integral \( \int e^{-\sqrt{x}} \, dx \) into \( 2 \int u e^{-u} \, du \).

The next step, integration by parts, provided a way to further simplify this integral. The key to successful evaluation is recognizing when and which technique to apply.
  • Start by analyzing the integral's structure.
  • Consider substitution if it can simplify the expression.
  • Opt for integration by parts if the expression is a product of functions.
  • Systematically simplify until you can integrate directly.
This systematic approach ensures you can tackle a wide variety of integrals with confidence.
Computer Algebra System (CAS)
A Computer Algebra System (CAS) is a software tool that facilitates symbolic mathematics. This includes differentiation, integration, solving equations, and more. These systems greatly enhance our ability to evaluate complex integrals efficiently.
They make checking our manual calculations straightforward. In the exercise, after manually solving the integral, we used a CAS to verify the solution.
  • Enter the original integral into the CAS.
  • Observe the CAS's symbolic computation.
  • Compare the CAS output with the manually found result.
By ensuring both results match, we confirm the mathematical correctness of our manual method. Using CAS not only checks correctness but also offers insight into potentially more optimal solutions.
This showcases its value beyond mere computation.

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Most popular questions from this chapter

Medication can be administered to a patient using a variety of methods. For a given method, let \(c(t)\) denote the concentration of medication in the patient's bloodstream (measured in \(\mathrm{mg} / \mathrm{L}\) ) \(t\) hours after the dose is given. The area under the curve \(c=c(t)\) over the time interval \([0,+\infty)\) indicates the "availability" of the medication for the patient's body. Determine which method provides the greater availability. Method \(1: c_{1}(t)=6\left(e^{-0.4 t}-e^{-1.3 t}\right)\) Method \(2: c_{2}(t)=5\left(e^{-0.4 t}-e^{-3 t}\right)\)

A transform is a formula that converts or "transforms" one function into another. Transforms are used in applications to convert a difficult problem into an easier problem whose solution can then be used to solve the original difficult problem. The Laplace transform of a function \(f(t),\) which plays an important role in the study of differential equations, is denoted by \(\mathscr{L}\\{f(t)\\}\) and is defined by $$\mathscr{L}\\{f(t)\\}=\int_{0}^{+\infty} e^{-s t} f(t) d t$$ In this formula \(s\) is treated as a constant in the integration process; thus, the Laplace transform has the effect of transforming \(f(t)\) into a function of \(s .\) Use this formula in these exercises. In each part, find the Laplace transform. (a) \(f(t)=t, s>0\) (b) \(f(t)=t^{2}, s>0\) (c) \(f(t)=\left\\{\begin{array}{ll}0, & t<3 \\ 1, & t \geq 3\end{array}, s>0\right.\)

Writing Explain how the product rule for derivatives and the technique of integration by parts are related.

Determine whether the statement is true or false. Explain your answer. To evaluate \(\int \sin ^{5} x \cos ^{8} x d x,\) use the trigonometric identity \(\sin ^{2} x=1-\cos ^{2} x\) and the substitution \(u=\cos x\)

Evaluate \(\int x \tan ^{-1} x d x\) using integration by parts. Simplify the computation of \(\int v \, d u\) by introducing a constant of integration \(C_{1}=\frac{1}{2}\) when going from \(d v\) to \(v\)
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