Problem 21
Question
Use the table of integrals at the back of the book to evaluate the integrals. \(\int e^{2 t} \cos 3 t d t\)
Step-by-Step Solution
Verified Answer
\( \int e^{2t} \cos 3t \, dt = \frac{e^{2t}}{13} (2 \cos(3t) + 3 \sin(3t)) + C \).
1Step 1: Identify the Integral Type
The given integral is of the form \( \int e^{a t} \cos(b t)\, dt \). This is a standard form typically found in tables of integrals.
2Step 2: Locate the Integral Formula
Look for a table of integrals in your textbook, specifically for \( \int e^{a t} \cos(b t)\, dt \). The formula is generally \[ \int e^{a t} \cos(b t)\, dt = \frac{e^{a t}}{a^2 + b^2} (a \cos(b t) + b \sin(b t)) + C. \]
3Step 3: Substitute Values
Substitute \( a = 2 \) and \( b = 3 \) into the formula: \[ \int e^{2 t} \cos 3 t \, dt = \frac{e^{2 t}}{2^2 + 3^2} (2 \cos(3 t) + 3 \sin(3 t)) + C. \]
4Step 4: Simplify the Denominator
Calculate the expression in the denominator: \( a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 \).
5Step 5: Write the Final Answer
Replace the denominator in the formula with 13: \[ \int e^{2 t} \cos 3 t \, dt = \frac{e^{2 t}}{13} (2 \cos(3 t) + 3 \sin(3 t)) + C. \]
Key Concepts
Table of IntegralsExponential FunctionsTrigonometric Integration
Table of Integrals
A table of integrals is like a cheat sheet for students working on calculus problems. It contains a list of integral formulas, which makes solving complex integrals much faster and easier. These tables are especially useful when dealing with integrals that don't have obvious solutions. Typically, they include standard forms for functions like polynomials, exponentials, and trigonometric functions.
Look at your specific case, the integral \( \int e^{2 t} \cos 3 t \, dt \), where the table helps identify quickly that this integral fits a standard form found in the table: \( \int e^{a t} \cos(b t)\, dt \).
Using the provided formula in the table allows you to substitute known values for easy evaluation, turning complex operations into straightforward arithmetic.
Look at your specific case, the integral \( \int e^{2 t} \cos 3 t \, dt \), where the table helps identify quickly that this integral fits a standard form found in the table: \( \int e^{a t} \cos(b t)\, dt \).
Using the provided formula in the table allows you to substitute known values for easy evaluation, turning complex operations into straightforward arithmetic.
Exponential Functions
Exponential functions are a fundamental part of calculus due to their unique properties. An exponential function has a variable in the exponent, like \( e^{x} \). They grow or decay at rates proportional to their current value, which is why they appear in so many natural processes.
In the given integral \( \int e^{2 t} \cos 3 t \), the exponential part \( e^{2t} \) shows how rapidly something can grow or shrink depending on the context.
Using exponential functions in integration often involves combining them with other function types, a typical scenario in solving differential equations or modeling physical situations involving growth, decay, or wave-like behavior.
In the given integral \( \int e^{2 t} \cos 3 t \), the exponential part \( e^{2t} \) shows how rapidly something can grow or shrink depending on the context.
Using exponential functions in integration often involves combining them with other function types, a typical scenario in solving differential equations or modeling physical situations involving growth, decay, or wave-like behavior.
Trigonometric Integration
Trigonometric functions are also commonly found in integrals. Functions like \( \sin \) and \( \cos \) are periodic, making them ideal for modeling wave motions, oscillations, and rotations. Integrating these functions can sometimes be straightforward, but when combined with other types, like exponentials, it gets a bit more challenging.
For example, in \( \int e^{2 t} \cos 3 t \, dt \), you need to use a combination of both the exponential and trigonometric integration methods. The integration table provides a formula tailored for these combinations, which simplifies the process of working out their integral.
Mastering trigonometric integration is key to solving many real-world applications like electrical engineering and signal processing.
For example, in \( \int e^{2 t} \cos 3 t \, dt \), you need to use a combination of both the exponential and trigonometric integration methods. The integration table provides a formula tailored for these combinations, which simplifies the process of working out their integral.
Mastering trigonometric integration is key to solving many real-world applications like electrical engineering and signal processing.
Other exercises in this chapter
Problem 20
Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int \frac{e^{\sqrt{t}} d t}{\sqrt{t}} $$
View solution Problem 21
Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{d x}{\left(x^{2}-1\right)^{3 / 2}}, \quad x>1 $$
View solution Problem 21
Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{-\infty}^{0} \theta e^{\theta} d \theta $$
View solution Problem 21
Evaluate the integrals in Exercises \(15-22\). $$ \int_{0}^{\pi / 2} \theta \sqrt{1-\cos 2 \theta} d \theta $$
View solution