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Chapter 7: Problem 24

Evaluate the following integrals. $$\int e^{3 x} \cos 2 x d x$$

Short Answer

Expert verified
Question: Find the integral of the function $$\int e^{3x}\cos{2x} dx.$$ Answer: The integral of the function is $$\int e^{3x}\cos{2x} dx = \frac{4}{13}e^{3x}\sin{2x} + \frac{3}{13}e^{3x}\cos{2x} + C,$$ where C is the constant of integration.

Step by step solution

01

Integration by Parts

Recall the integration by parts formula: $$\int u dv = uv - \int v du.$$ Let $$u = e^{3x}$$ and $$dv = \cos{2x} dx.$$ Therefore, $$du = 3e^{3x} dx$$ and $$v = \int \cos{2x} dx = \frac{1}{2}\sin{2x}.$$ Apply the integration by parts formula: $$\int e^{3x}\cos{2x} dx = e^{3x} \left(\frac{1}{2}\sin{2x}\right) - \int \left(\frac{1}{2}\sin{2x}\right) (3e^{3x}) dx.$$
02

Integration by Parts Again

Now, let $$u = 3e^{3x}$$ and $$dv = \frac{1}{2}\sin{2x} dx.$$ Therefore, $$du = 9e^{3x} dx$$ and $$v = \int \frac{1}{2}\sin{2x} dx = -\frac{1}{4}\cos{2x}.$$ Apply the integration by parts formula again: $$\int 3e^{3x}\frac{1}{2}\sin{2x} dx = 3e^{3x} \left(-\frac{1}{4}\cos{2x}\right) - \int \left(-\frac{1}{4}\cos{2x}\right) (9e^{3x}) dx.$$
03

Simplify

Simplify the expression: $$\int e^{3x}\cos{2x} dx = \frac{1}{2}e^{3x}\sin{2x} + \frac{3}{4}e^{3x}\cos{2x} - \frac{9}{4}\int e^{3x}\cos{2x} dx.$$
04

Solve for the Original Integral

Notice that we have the original integral on the right side. Now, we will solve for $$\int e^{3x}\cos{2x} dx$$: $$\frac{13}{4}\int e^{3x}\cos{2x} dx = \frac{1}{2}e^{3x}\sin{2x} + \frac{3}{4}e^{3x}\cos{2x} \Rightarrow \int e^{3x}\cos{2x} dx = \frac{4}{13}e^{3x}\sin{2x} + \frac{3}{13}e^{3x}\cos{2x} + C,$$ where C is the constant of integration.

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Key Concepts

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

Definite Integrals
While the original exercise is concerned with an indefinite integral, understanding definite integrals is key to mastering integrals in general. A definite integral calculates the area under the curve of a function between two specific points. It's denoted as \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the lower and upper limits, respectively.
In finding the area, the definite integral considers not only the values of the function but also the distance between \(a\) and \(b\).
  • If \(f(x)\) is positive in the interval \([a, b]\), the area adds to the total sum.
  • If \(f(x)\) is negative, the area subtracts from the sum.
  • For complex functions like exponential or trigonometric functions, breaking the problem down can be necessary.
Using substitution or integration by parts can ease the process of calculating definite integrals, just as it does with indefinite integrals.
Exponential Functions
Exponential functions, like \(e^{3x}\) in the given problem, are crucial in many areas of mathematics and sciences. The key feature of an exponential function is its constant growth rate, which means it grows or shrinks proportionally depending on the exponent.
Here are some characteristics of exponential functions that can help in integration:
  • Exponential functions are their own derivatives and integrals, which means \(\frac{d}{dx}e^{kx} = ke^{kx}\).
  • When combined with other functions, like trigonometric functions, exponential functions require methods like integration by parts for solving integrals.
  • Exponential functions tend to dominate other types like polynomial or logarithmic functions when both are mixed in an expression.
In the integration example, \(e^{3x}\) is used in combination with trigonometric functions, highlighting the need for advanced integration methods.
Trigonometric Functions
Trigonometric functions, such as \(\cos(2x)\) in the exercise, oscillate between set values, making them unique and interesting in calculus. Here are some key concepts:
  • Trigonometric functions have well-known derivatives and integrals that facilitate their use in calculus. For example, \(\int \cos(ax) \, dx = \frac{1}{a}\sin(ax) + C\).
  • Because of their periodic nature, trigonometric functions often require specialized techniques when integrated, especially when combined with exponential functions.
  • In this particular exercise, recognizing the integral of \(\cos(2x)\) is critical to applying the integration by parts method.
These functions, when paired with others like exponentials, necessitate strategic methods, as seen in this integral solution, to process their combined effects properly.

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Most popular questions from this chapter

Use integration by parts to evaluate the following integrals. $$\int_{1}^{\infty} \frac{\ln x}{x^{2}} d x$$

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t)\) the Laplace transform is a new function \(F(s)\) defined by $$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$ where we assume that \(s\) is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1}$$ Verify the following Laplace transforms, where a is a real number. $$f(t)=e^{a t} \longrightarrow F(s)=\frac{1}{s-a}$$

Solve the following problems using the method of your choice. $$\frac{d z}{d x}=\frac{z^{2}}{1+x^{2}}, z(0)=\frac{1}{6}$$

The Eiffel Tower property Let \(R\) be the region between the curves \(y=e^{-\alpha x}\) and \(y=-e^{-\alpha x}\) on the interval \([a, \infty),\) where \(a \geq 0\) and \(c>0 .\) The center of mass of \(R\) is located at \((\bar{x}, 0)\) where \(\bar{x}=\frac{\int_{a}^{\infty} x e^{-c x} d x}{\int_{a}^{\infty} e^{-c x} d x} .\) (The profile of the Eiffel Tower is modeled by the two exponential curves; see the Guided Project The exponential Eiffel Tower.) a. For \(a=0\) and \(c=2,\) sketch the curves that define \(R\) and find the center of mass of \(R\). Indicate the location of the center of mass. b. With \(a=0\) and \(c=2,\) find equations of the lines tangent to the curves at the points corresponding to \(x=0\) c. Show that the tangent lines intersect at the center of mass. d. Show that this same property holds for any \(a \geq 0\) and any \(c>0 ;\) that is, the tangent lines to the curves \(y=\pm e^{-c x}\) at \(x=a\) intersect at the center of mass of \(R\).

Prove that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.
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