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Magazine Chapter 28: Problem 16
Integrate each of the given functions. $$\int_{0}^{1} \frac{2 e^{s} d s}{\sec e^{s}}$$
Short Answer
Expert verified
The integral evaluates to \( 2 \sin(e) - 2 \sin(1) \).
Step by step solution
01
Simplify the Integrand
The function we are given to integrate is \( \frac{2 e^s}{\sec(e^s)} \). We know that \( \sec(x) = \frac{1}{\cos(x)} \). Therefore, \( \frac{1}{\sec(x)} = \cos(x) \). We can simplify the integrand as follows: \( 2 e^s \cos(e^s) \).
02
Setup the Integral with the New Integrand
Replacing the original integrand with the simplified expression, we rewrite the integral: \( \int_{0}^{1} 2 e^s \cos(e^s) \; ds \).
03
Choose a Substitution
Notice that the function \( e^s \) appears both in the argument of the cosine and as a factor in the integrand. This suggests that a substitution might simplify the integration process. Let's use the substitution \( u = e^s \), which gives \( du = e^s \; ds \) or equivalently \( ds = \frac{du}{e^s} = \frac{du}{u} \).
04
Change the Limits of Integration
With the substitution \( u = e^s \), when \( s = 0 \), \( u = e^0 = 1 \), and when \( s = 1 \), \( u = e^1 = e \). Thus, the limits of integration change from \( s = 0 \) to \( s = 1 \) into \( u = 1 \) to \( u = e \).
05
Rewrite the Integral in Terms of \( u \)
The integral now becomes: \( \int_{1}^{e} 2 \cos(u) \; du \). The factor \( e^s \) in the numerator has been replaced by \( u \), and thus has no influence on the restructured integrand.
06
Integrate \( 2 \cos(u) \) with Respect to \( u \)
The antiderivative of \( \cos(u) \) is \( \sin(u) \). Therefore, integrating \( 2 \cos(u) \) gives \( 2 \sin(u) \).
07
Evaluate the Definite Integral
Substitute the limits of integration into the antiderivative: \( 2 \sin(u) \bigg|_{1}^{e} \). This evaluates to \( 2 \sin(e) - 2 \sin(1) \).
08
Conclude with the Final Result
The final answer after evaluating the expression is \( 2 \sin(e) - 2 \sin(1) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
The concept of a definite integral is foundational in calculus. It represents the accumulation of quantities, such as area under a curve, over a specific interval. In our original exercise, we worked with the definite integral \[ \int_{0}^{1} \frac{2 e^{s}}{\sec(e^{s})} ds \].By definition, a definite integral has limits of integration, in this case, from 0 to 1. These limits tell us where to start and where to stop accumulating area under the curve.
- Start at the lower limit and add up slices until you reach the upper limit.
- This provides the net area, considering positive and negative values.
Substitution Method
The substitution method is a powerful technique in integration. It simplifies an integral by changing the variable of integration, often making the integral easier to solve. In our problem, we used the substitution \( u = e^{s} \).
- By letting \( u = e^{s} \), it becomes manageable.
- Compute \( du = e^s \, ds \), leading to \( ds = \frac{du}{e^s} = \frac{du}{u} \).
Trigonometric Integration
Trigonometric integration deals with integrals involving trigonometric functions. In this exercise, we transformed \( \frac{2e^s}{\sec(e^s)} \) into \( 2e^s \cos(e^s) \).
- The simplification uses the fact that \( \frac{1}{\sec(x)} = \cos(x) \).
- By converting trigonometric forms, the integral becomes straightforward.
Antiderivative
An antiderivative is the reverse operation of differentiation. To solve our problem, we find the antiderivative of \( 2 \cos(u) \).
- The antiderivative of \( \cos(u) \) is \( \sin(u) \).
- So the antiderivative of \( 2 \cos(u) \) is \( 2 \sin(u) \).
