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Magazine Chapter 28: Problem 17
In Exercises \(3-26,\) integrate each of the given functions. $$\int_{0}^{\pi / 2} e^{x} \cos x d x$$
Short Answer
Expert verified
The integral evaluates to \(\frac{e^{\pi/2}}{2} - \frac{1}{2}\).
Step by step solution
01
Set Up Integration by Parts Formula
The integration by parts formula is \(\int u \, dv = uv - \int v \, du\). In this step, we identify parts of the integrand: let \(u = e^x\) and \(dv = \cos x\, dx\).
02
Differentiate and Integrate Functions
Differentiate \(u\) to find \(du\) and integrate \(dv\) to find \(v\). We get \(du = e^x \, dx\) and \(v = \sin x\).
03
Substitute into Integration by Parts Formula
Substitute \(u, v, du,\) and \(dv\) into the integration by parts formula: \(\int e^x \cos x \, dx = e^x \sin x - \int e^x \sin x \, dx\). This creates a new integral.
04
Apply Integration by Parts Again
For the new integral \(\int e^x \sin x \, dx\), use integration by parts again. Let \(u = e^x\) and \(dv = \sin x \, dx\), giving \(du = e^x \, dx\) and \(v = -\cos x\).
05
Substitute and Simplify
Applying integration by parts again: \(\int e^x \sin x \, dx = -e^x \cos x - \int (-e^x \cos x) \, dx\). Simplify to get \(-e^x \cos x + \int e^x \cos x \, dx\).
06
Solve the Equation System
Substitute back to original equation: \(\int e^x \cos x \, dx = e^x \sin x + e^x \cos x - \int e^x \cos x \, dx\). Add \(\int e^x \cos x \, dx\) to both sides to solve, yielding \(2 \int e^x \cos x \, dx = e^x (\sin x + \cos x)\).
07
Dividing by Two
Divide both sides by 2 to find \(\int e^x \cos x \, dx = \frac{e^x}{2} (\sin x + \cos x)\).
08
Evaluate the Definite Integral
Evaluate from 0 to \(\pi/2\): \[ \left[ \frac{e^x}{2}(\sin x + \cos x) \right]_0^{\pi/2} = \frac{e^{\pi/2}}{2} - \frac{1}{2} \].
09
Calculate Final Value
Calculate the definite values: \(\frac{e^{\pi/2}}{2} \cdot (1+0) - \frac{1}{2} \cdot (0+1) = \frac{e^{\pi/2}}{2} - \frac{1}{2}\).
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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 useful technique in calculus. It helps find the integral of products of functions.
To apply it, we use the formula: - \( \int u \, dv = uv - \int v \, du \)Here, we split the function into two parts: \( u \) and \( dv \).
This method breaks down a tough integral into simpler parts that we can solve step by step.
To apply it, we use the formula: - \( \int u \, dv = uv - \int v \, du \)Here, we split the function into two parts: \( u \) and \( dv \).
- Choose \( u \) so that it becomes simpler when differentiated.
- Choose \( dv \) so that it鈥檚 easy to integrate.
This method breaks down a tough integral into simpler parts that we can solve step by step.
Exponential Functions
Exponential functions, like \( e^x \), are functions where the variable appears in the exponent. They play a crucial role in calculus due to their unique properties:
They grow quickly, which is why \( e^x \) is called the natural exponential function. However, when combined with other functions, like trigonometric functions, they require techniques like integration by parts to solve their integrals.
In the given exercise, \( e^x \) easily fits into the integration by parts formula due to its simplicity in differentiation.
- The derivative of \( e^x \) is \( e^x \).
- The integral of \( e^x \) is \( e^x + C \).
They grow quickly, which is why \( e^x \) is called the natural exponential function. However, when combined with other functions, like trigonometric functions, they require techniques like integration by parts to solve their integrals.
In the given exercise, \( e^x \) easily fits into the integration by parts formula due to its simplicity in differentiation.
Trigonometric Functions
Trigonometric functions, such as \( \sin x \) and \( \cos x \), are based on the angles of a triangle. They are periodic, which means they repeat at regular intervals. Some important properties include:
When paired with exponential functions, as seen in this exercise, they might seem tough to integrate. Integration by parts is the go-to method here.In this exercise, we use \( \cos x \) and \( \sin x \) in combination with \( e^x \).
This requires two rounds of integration by parts to reduce the problem into a simpler form.
- \( \sin x \) and \( \cos x \) have derivatives \( \cos x \) and \(-\sin x \), respectively.
- Integrals: \( \int \sin x \, dx = -\cos x + C \) and \( \int \cos x \, dx = \sin x + C \).
When paired with exponential functions, as seen in this exercise, they might seem tough to integrate. Integration by parts is the go-to method here.In this exercise, we use \( \cos x \) and \( \sin x \) in combination with \( e^x \).
This requires two rounds of integration by parts to reduce the problem into a simpler form.
