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Chapter 6: Problem 23

Evaluate the definite integrals. \(\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x\)

Short Answer

Expert verified
The definite integral evaluates to \( 2(e^2 - e) \).

Step by step solution

01

Identify the Type of Integral

Notice that the integral \( \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx \) involves an exponential function and a fractional power in its denominator. This suggests that it might be suitable for a substitution method.
02

Choose Appropriate Substitution

Use the substitution \( u = \sqrt{x} \), which implies \( x = u^2 \). Differentiate \( x = u^2 \) to find \( dx \); thus, \( dx = 2u \, du \).
03

Change the Limits of Integration

Since \( u = \sqrt{x} \), when \( x = 1 \), \( u = \sqrt{1} = 1 \), and when \( x = 4 \), \( u = \sqrt{4} = 2 \). So the limits of integration change from \( x = 1 \) and \( x = 4 \) to \( u = 1 \) and \( u = 2 \).
04

Substitute into the Integral

Substitute \( u = \sqrt{x} \) and \( dx = 2u \, du \) into the integral. The integral becomes: \[\int_{1}^{2} \frac{e^u}{u} \cdot 2u \, du = 2 \int_{1}^{2} e^u \, du\]
05

Evaluate the Integral

The antiderivative of \( e^u \) is simply \( e^u \). Thus, the integral becomes:\[2 \left[ e^u \right]_{1}^{2}\]
06

Apply the Limits of Integration

Substitute the limits of integration into the antiderivative:\[2(e^2 - e^1) = 2(e^2 - e)\]
07

Final Calculation

Calculate the expression \( 2(e^2 - e) \) to find the numerical value if needed or leave it in this simplified form.

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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 technique used in calculus to simplify the process of solving integrals. It involves changing variables to transform a more complicated integral into a simpler one. In our exercise, we encounter an integral with an exponential function and a fractional power,which suggests the use of substitution could be beneficial.
  • We choose to substitute the expression inside the exponential function. Here, we set \( u = \sqrt{x} \), which means \( x = u^2 \).
  • This substitution simplifies the expression because differentiating \( x = u^2 \) gives us the derivative \( dx = 2u \, du \).
  • Replacing these into the integral helps us simplify the integration, as it transforms our original integral into a more manageable form.
This technique is particularly useful when dealing with integrals involving compositions of functions, making seemingly complex integrals easier to handle.
Exponential Function
An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. The exponential function \( e^x \) is one of the most important functions in mathematics. Its unique property is that it is equal to its own derivative, meaning the rate of change of \( e^x \) is \( e^x \).
  • In the exercise, the integral's numerator is \( e^{\sqrt{x}} \), exemplifying this function.
  • The process of integration requires us to find the antiderivative of this exponential expression.
  • After substitution, the exponential becomes \( e^u \), which is straightforward to integrate given its antiderivative remains \( e^u \).
Understanding exponential functions is crucial in solving problems in calculus, especially when they are coupled with other functions like polynomials or trigonometric functions.
Integration Limits
The integration limits specify the range over which an integral is evaluated. They are crucial for computing definite integrals, which provide a numerical value representing the area under a curve between two points.
  • In the substitution process, changing variables also requires transforming these limits to correspond to the new variable.
  • Initially, the limits of our integral are from \( x = 1 \) to \( x = 4 \).
  • After substituting \( u = \sqrt{x} \), we compute the new limits: when \( x=1 \), \( u=1 \); and when \( x=4 \), \( u=2 \).
These new limits ensure the simplified integral accurately represents the same area under the curve as the original problem. Properly adjusting integration limits is a key step in the substitution method.
Antiderivative
Finding the antiderivative is a fundamental part of solving integrals. It involves determining a function whose derivative is the integrand (the expression inside the integral).
  • In our problem, after substitution, the integral reduces to \( 2 \int_{1}^{2} e^u \, du \).
  • The antiderivative of \( e^u \) is simply \( e^u \), reflecting the property of the exponential function.
  • Substituting back the new limits into the evaluated antiderivative gives the result: \( 2[e^u]_{1}^{2} = 2(e^2 - e) \).
Antiderivatives are essential for evaluating definite integrals, offering insight into how functions accumulate area or change across intervals.

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Most popular questions from this chapter

In Exercises 67 and 68 of Section 3.3 we noted that \(\frac{d}{d x}(\sin x)=\cos x\) and \(\frac{d}{d x}(\cos x)=-\sin x .\) Using this, find a. \(\int \sin x d x\) b. \(\int \cos x d x\)

In Exercises 1 through 38 , find the antiderivatives. $$ \int \frac{x+1}{x} d x $$

Use the order properties of the definite integral to establish the inequalities. $$ 3 \leq \int_{1}^{4} \sqrt{x} d x \leq 6 $$

A firm has found that its costs in billions of dollars have been increasing (owing primarily to inflation) at a rate given by \(C_{1}^{\prime}(t)=1.2 t,\) where \(t\) is the time measured in years since production of the item began. At the beginning of the second year an innovation in the production process resulted in costs increasing at a rate given by \(C_{2}^{\prime}(t)=0.9 \sqrt{t}\). Find the total costs during the first 4 years. Assume \(C_{1}(0)=0 .\)

In Exercises 1 through 38 , find the antiderivatives. $$ \int\left(1+\frac{3}{x}\right) d x $$
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