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Chapter 8: Problem 22

Use the table of integrals at the back of the book to evaluate the integrals. \(\int e^{-3 t} \sin 4 t d t\)

Short Answer

Expert verified
\( \int e^{-3t} \sin 4t \, dt = \frac{e^{-3t}}{25} (-3 \sin(4t) - 4 \cos(4t)) + C \).

Step by step solution

01

Recognize the Integral Type

Notice that the integral \( \int e^{-3t} \sin 4t \, dt \) is of the form \( \int e^{at} \sin(bt) \, dt \). This matches a common form found in most integral tables.
02

Identify the Appropriate Formula

In the integral table, find the formula for \( \int e^{ax} \sin(bx) \, dx \), usually given as: \[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C \]. Here, \( a = -3 \) and \( b = 4 \).
03

Substitute the Appropriate Constants

Plug \( a = -3 \) and \( b = 4 \) into the formula: \[ \frac{e^{-3t}}{(-3)^2 + 4^2} ((-3) \sin(4t) - 4 \cos(4t)) + C \].
04

Simplify the Expression

Calculate \( (-3)^2 + 4^2 = 9 + 16 = 25 \) and simplify the expression: \[ \frac{e^{-3t}}{25} (-3 \sin(4t) - 4 \cos(4t)) + C \].
05

Write the Final Result

Therefore, the evaluated integral is: \[ \int e^{-3t} \sin 4t \, dt = \frac{e^{-3t}}{25} (-3 \sin(4t) - 4 \cos(4t)) + C \].

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Key Concepts

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

Integration by Formula
In integral calculus, one efficient method for finding integrals is the use of integration formulas. Integration by formula simplifies the integration process by using established formulas for specific types of integrals. This is especially helpful where direct integration may involve complex steps or advanced techniques. When faced with an integral, the first step is to identify it in a form resembling one of these known formulas.
  • Integration tables list standard forms, allowing you to substitute variables and constants directly to obtain the integral.
  • These forms are derived from fundamental integration rules, which make them trustworthy for obtaining accurate results swiftly.
Integration by formula thus acts as a shortcut, but it requires practice to recognize patterns and appropriate applications. In the provided exercise, recognizing the integral as matching a standard form simplifies the problem significantly.
Exponential Functions
Exponential functions play a crucial role in various fields of mathematics and science. In general, exponential functions have the form \(e^{kx}\), where \(e\) is Euler's number (approximately 2.71828), and \(k\) is a constant. They exhibit continuous growth or decay, depending on the value of \(k\).
  • In integrals, exponential functions often appear when modeling phenomena such as radioactive decay, population growth, and certain economic models.
  • The derivative and the integral of \(e^{kx}\) maintain similar forms, making them quite manageable in calculus.
In our exercise, we deal with the function \(e^{-3t}\), highlighting exponential decay. When integrating exponential functions in conjunction with other functions, such as sines and cosines, formula-based integration can greatly assist in simplifying the process.
Trigonometric Integrals
Trigonometric integrals involve integrating expressions containing trigonometric functions like sine, cosine, tangent, etc. These integrals are crucial when dealing with periodic functions and oscillatory behavior in physics and engineering.
  • Common trigonometric functions include \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), and others, often forming part of more complex integrals.
  • Understanding the properties, such as periodicity and symmetry, of these functions can be very useful when solving integrals.
    • For the exercise in question, we used the sine function, which represents an oscillating behavior. Trigonometric integrals often appear together with exponential functions, as seen in our example, yielding integrals commonly found in signal processing and other oscillatory phenomena. Recognizing these combined forms is key, making the use of integral tables highly beneficial for quick and accurate solutions.

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    Most popular questions from this chapter

    Refer to formulas in the table of integrals at the back of the book. Derive Formula 46 by using a trigonometric substitution to evaluate $$\int \frac{d x}{x^{2} \sqrt{x^{2}-a^{2}}}$$

    Use the integral tables to evaluate the integrals. \(\int \operatorname{sech}^{7} x \tanh x d x\)

    The error function The error function, $$ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t $$ important in probability and in the theories of heat flow and signal transmission, must be evaluated numerically because there is no elementary expression for the antiderivative of \(e^{-t^{2}} .\) a. Use Simpson's Rule with \(n=10\) to estimate erf \((1) .\) b. \(\ln [0,1]\) $$ \left|\frac{d^{4}}{d t^{4}}\left(e^{-t^{2}}\right)\right| \leq 12 $$ Give an upper bound for the magnitude of the error of the estimate in part (a).

    In Exercises \(11-14\) , use the tabulated values of the integrand to estimate the integral with (a) the Trapezoidal Rule and (b) Simpson's Rule with \(n=8\) steps. Round your answers to five decimal places. Then (c) find the integral's exact value and the approximation error \(E_{T}\) or \(E_{s}\) as appropriate. $$ \int_{\pi / 4}^{\pi / 2}\left(\csc ^{2} y\right) \sqrt{\cot y} d y $$

    Consider the integral \(\int_{-1}^{1} \sin \left(x^{2}\right) d x\) a. Find \(f^{(4)}\) for \(f(x)=\sin \left(x^{2}\right) .\) (You may want to check your work with a CAS if you have one available.) b. Graph \(y=f^{(4)}(x)\) in the viewing window \([-1,1]\) by \([-30,10] .\) c. Explain why the graph in part (b) suggests that \(\left|f^{(4)}(x)\right| \leq 30\) for \(-1 \leq x \leq 1\) d. Show that the error estimate for Simpson's Rule in this case becomes $$ \left|E_{S}\right| \leq \frac{(\Delta x)^{4}}{3} $$ e. Show that the Simpson's Rule error will be less than or equal to 0.01 in magnitude if \(\Delta x \leq 0.4 .\) f. How large must \(n\) be for \(\Delta x \leq 0.4 ?\)
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