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Chapter 6: Problem 12

Evaluate the integral. $$\int_{0}^{\pi / 4} \sec ^{2} x e^{\tan x} d x$$

Short Answer

Expert verified
The value of the integral is \( e - 1 \)

Step by step solution

01

Define Substitution

Define the substitution \(u=\tan(x)\). Then compute the derivative of \(u\) with respect to \(x\), which is \(du/\sec^2(x)dx\). When re-arranged, the differential term in the integral transforms into \(dx=du/\sec^2(x)\). This changes the integral into \( \int e^u du\), which is easier to solve.
02

Replace the Integration Interval

Replace the interval of integration from [0, 蟺/4] to [0, 1] due to the substitution \( u = \tan(x)\). Within this interval for \( x\), the tangent function increases from 0 to 1.
03

Perform the Integral

So the integral to be solved becomes \( \int_{0}^{1} e^u du \). The evaluation of this integral using standard rules results in \( [e^u]_{0}^{1} \), which simplifies down to \( e^1 - e^0 = e - 1 \)

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Key Concepts

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

U-Substitution
The technique known as u-substitution is a method used in integral calculus to simplify the process of integration, especially when dealing with more complicated expressions. It is particularly useful when an integrand is the product of a function and its derivative or can be manipulated to resemble such a form. The basic idea is to substitute a part of the integral (call it u) with a new variable, making the integral easier to evaluate.

To perform u-substitution, we typically follow these steps:
  • Identify a section of the integral to replace with u, ideally so that du is also present in the integral.
  • Compute du, the derivative of u with respect to x, and solve for dx.
  • Replace all occurrences of the chosen section with u and dx with du, thus rewriting the integral in terms of u.
  • Change the limits of integration if performing a definite integral to reflect the new variable u.
  • Evaluate the new integral.
  • Substitute back to the original variable if needed.
This approach transforms the original integral into one that is typically more straightforward to integrate. It is important to note that the chosen substitution must simplify the integral; a poor choice of u can lead to more complexity.
Definite Integral
The definite integral of a function provides us with the net area under the curve of that function between two points on the x-axis. These two points are referred to as the lower and upper limits of integration. The evaluation of a definite integral results in a real number that can represent physical quantities like area, volume, or other concepts depending on the context of the problem.

To evaluate a definite integral, we:
  • Find an antiderivative (also called the indefinite integral) of the function being integrated.
  • Apply the limits of integration to this antiderivative, which involves substituting these limits into the antiderivative.
  • Calculate the difference between the upper limit's substitution result and the lower limit's substitution result.
This process is often represented symbolically by the Fundamental Theorem of Calculus, which connects differentiation with integration and provides a powerful tool for evaluating definite integrals. In the case of our exercise involving u-substitution, after applying the new limits of integration, we find the definite integral in terms of u, making it much simpler to compute.
Exponential Functions
An exponential function in mathematics can be recognized by its base raised to a variable exponent, typically represented as f(x) = a^x, where a is a constant base and x is the exponent. One of the most important and widely used exponential functions is e^x, where e is Euler鈥檚 number approximately equal to 2.71828. This function is unique because it is its own derivative, which makes it incredibly useful in various fields, including calculus and real-world applications like compounding interest and population growth.

When integrating an exponential function, the general form of the antiderivative of e^x is also e^x, plus a constant of integration in the case of an indefinite integral. For a definite integral, the result is the difference in the values of e^x evaluated at the upper and lower limits of integration. In our exercise, after u-substitution, we integrate e^u over the range 0 to 1, leading us to simply find the difference e^1 - e^0, which neatly evaluates to e - 1.

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Most popular questions from this chapter

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In exercises find the partial fractions decomposition and an antiderivative. If you have a CAS available, use it to check your answer. $$\frac{1}{x^{3}+4 x}$$

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