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Chapter 5: Problem 2

Use integration by parts to evaluate the integral. \(\int 3 x \cos (2 x) d x=\) ___________

Short Answer

Expert verified
\begin{equation*} I = \frac {3}{2} x \sin(2x) + \frac {3}{4} \cos(2x) + C \end{equation*}

Step by step solution

01

- Identify Parts for Integration by Parts

For the integral \(\begin{equation*} \begin{aligned} I = \int 3x \cos(2x) dx \end{aligned} \end{equation*} \), choose \(u = 3x\) and \(dv = \cos(2x) dx\).
02

- Differentiate and Integrate

Find \( du \) by differentiating \( u \): \(\begin{equation*} \begin{aligned} du = 3 dx \end{aligned} \end{equation*} \). Find \(v\) by integrating \(dv\): \(\begin{equation*} \begin{aligned} v = \int \cos(2x) dx = \frac {1}{2} \sin(2x) \end{aligned} \end{equation*} \).
03

- Apply Integration by Parts Formula

Apply the formula \(\begin{equation*} \begin{aligned} \int u dv = uv - \int v du \end{aligned} \end{equation*} \) to get: \(\begin{equation*} \begin{aligned} I = 3x \left(\frac {1}{2}\sin(2x)\right) - \int \frac {1}{2}\sin(2x) \left(3dx\right) \end{aligned} \end{equation*} \).
04

- Simplify and Evaluate Remaining Integral

Simplify: \(\begin{equation*} \begin{aligned} I = \frac {3}{2} x \sin(2x) - \frac {3}{2} \int \sin(2x) dx \end{aligned} \end{equation*} \). Next, evaluate the remaining integral: \(\begin{equation*} \begin{aligned} \frac {3}{2} \int \sin(2x) dx = \frac {3}{2} \left(- \frac {1}{2} \cos(2x)\right) = - \frac {3}{4} \cos(2x) \end{aligned} \end{equation*} \)
05

- Combine Terms and Add Constant of Integration

Combine the terms: \(\begin{equation*} \begin{aligned} I = \frac {3}{2} x \sin(2x) + \frac {3}{4} \cos(2x) + C \end{aligned} \end{equation*} \), 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.

Calculus
Calculus is a branch of mathematics that studies how things change. It's split into two main parts: differential calculus and integral calculus. Differential calculus focuses on how things change momentarily, studying rates of change and slopes of curves. Integral calculus sums up these rates of change over time, dealing with the accumulation of quantities and areas under curves. In solving integration problems such as the given exercise, concepts from both branches are often used.
Definite and Indefinite Integrals
Integrals are a key part of calculus. They come in two types: definite and indefinite.

Indefinite integrals, or antiderivatives, represent a family of functions whose derivative is the integrand. These integrals include a constant of integration (often denoted as C). For example, the indefinite integral of \(\frac{dx}{x}\) is \(\frac{ln|x| + C\).

Definite integrals, on the other hand, are calculated over a specific interval \([a, b]\). They represent the exact area under the curve between two points. For example, \(\frac{dx}{x}\) from 1 to 2 is \(\frac{ln|2|} - \frac{ln|1|}\).

In the given problem, we deal with an indefinite integral, aiming to find a general solution by applying the appropriate techniques.
Integration Techniques
Various techniques can be used to evaluate integrals. Here, we focus on integration by parts.

Integration by parts is based on the product rule for differentiation and follows the formula \(\frac{\text{鈭� u dv = uv - 鈭� v du}}\). This method is especially useful when dealing with products of functions, as seen in the given exercise.

For our integral, \(\frac{\text{鈭� 3x \text{cos(2x)} dx}}\), we choose \(\text{u = 3x}\) and \(\text{dv = cos(2x) dx}\).

1. Differentiate \(\text{u = 3x}\) to get \(\text{du = 3 dx}\).
2. Integrate \(\text{dv = cos(2x) dx}\) to obtain \(\text{v = (1/2) sin(2x)}\).

Applying the integration by parts formula gives: \(\frac{\text{I = 3x (1/2) sin(2x) - 鈭� v du}}\). This simplifies our integral evaluation. The process involves breaking down complex integrals into more manageable parts, using differentiation and integration systematically.

Remember, mastering these techniques requires practice and understanding the underlying principles. Happy integrating!

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Most popular questions from this chapter

Evaluate the definite integral. \(\int_{0}^{4} t e^{-t} d t=\) __________

For each of the following integrals involving rational functions, (1) use a CAS to find the partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting function without the assistance of technology; (3) use a CAS to evaluate the original integral to test and compare your result in (2). a. \(\int \frac{x^{3}+x+1}{x^{4}-1} d x\) b. \(\int \frac{x^{5}+x^{2}+3}{x^{3}-6 x^{2}+11 x-6} d x\) c. \(\int \frac{x^{2}-x-1}{(x-3)^{3}} d x\)

For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate, or if neither method is appropriate. Do not evaluate the integrals. 1\. \(\int x \sin x d x\) 2\. \(\int \frac{x^{2}}{1+x^{3}} d x\) 3\. \(\int x^{2} e^{x^{3}} d x\) 4\. \(\int x^{2} \cos \left(x^{3}\right) d x\) 5\. \(\int \frac{1}{\sqrt{3 x+1}} d x\) (Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.)

Consider the indefinite integral given by $$ \int \frac{\sqrt{x+\sqrt{1+x^{2}}}}{x} d x $$ a. Explain why \(u\) -substitution does not offer a way to simplify this integral by discussing at least two different options you might try for \(u\). b. Explain why integration by parts does not seem to be a reasonable way to proceed, either, by considering one option for \(u\) and \(d v\). c. Is there any line in the integral table in Appendix Athat is helpful for this integral? d. Evaluate the given integral using WolframAlpha. What do you observe?

The rate at which water flows through Table Rock Dam on the White River in Branson, MO, is measured in thousands of cubic feet per second (TCFS). As engineers open the floodgates, flow rates are recorded according to the following chart. $$ \begin{array}{llllllll} \hline \text { seconds, } t & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline \text { flow in TCFS, } r(t) & 2000 & 2100 & 2400 & 3000 & 3900 & 5100 & 6500 \\ \hline \end{array} $$ a. What definite integral measures the total volume of water to flow through the dam in the 60 second time period provided by the table above? b. Use the given data to calculate \(M_{n}\) for the largest possible value of \(n\) to approximate the integral you stated in (a). Do you think \(M_{n}\) over- or under-estimates the exact value of the integral? Why? c. Approximate the integral stated in (a) by calculating \(S_{n}\) for the largest possible value of \(n,\) based on the given data. d. Compute \(\frac{1}{60} S_{n}\) and \(\frac{2000+2100+2400+3000+3900+5100+6500}{7} .\) What quantity do both of these values estimate? Which is a more accurate approximation?
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